PX2132 Introductory Quantum Mechanics Notes (PDF)

PX2132: Introductory Quantum Mechanics F. Flicker August 18, 2023 These notes accompany the course videos. Details of derivations will appear here, but the videos convey the course and should be watched for surrounding explanation. Contents 1 The Motivation for Quantum Mechanics 5 1.1 The Schrödinger equation................................ 6 1.1.1 The time-dependent Schrödinger equation (TDSE).............. 6 1.1.2 The wavefunction ψ............................... 6 1.1.3 Assumptions going into the TDSE....................... 7 1.1.4 The Hamiltonian operator............................ 7 1.1.5 Separation of variables.............................. 8 1.2 How to Solve the Schrödinger Equation........................ 9 1.3 General boundary conditions.............................. 10 1.4 Probability density.................................... 11 1.5 Probability current density............................... 11 2 Scattering and tunnelling 13 2.1 TISE: solutions in regions of constant potential.................... 14 2.2 Plane waves........................................ 14 2.3 Scattering from a potential step............................. 15 2.3.1 Region I...................................... 15 2.3.2 Region II: E > V0................................ 16 2.3.3 Region II: E < V0................................ 17 2.3.4 Probability fluxes................................. 18 2.4 Quantum tunnelling................................... 20 2.4.1 E > V0...................................... 21 2.4.2 E < V0...................................... 22 1 2 PX2132: Introductory Quantum Mechanics 3 Bound states 23 3.1 The 1D infinite potential well.............................. 24 3.2 Normalisation....................................... 25 3.3 The Born Rule...................................... 26 3.4 The finite potential well................................. 26 4 Quantum Superposition 30 4.1 Stationary states..................................... 31 4.2 Quantum superposition................................. 31 4.3 Complete orthonormal bases............................... 32 4.4 Decomposing an arbitrary function into energy eigenstates.............. 32 4.5 Finding the subsequent time evolution of a given wavefunction........... 33 4.6 Measurement....................................... 34 4.7 Expectation values.................................... 35 5 Finite-dimensional Hilbert spaces 37 5.1 Complex vectors and matrices.............................. 38 5.2 Hermitian matrices.................................... 40 5.3 Spin-1/2.......................................... 42 5.3.1 Mathematical structure............................. 42 5.3.2 Measurements.................................. 44 5.3.3 The quantum eraser............................... 46 5.3.4 Expectation values................................ 47 6 Matrix Mechanics 49 6.1 Operators and observables................................ 51 6.2 The Heisenberg Uncertainty Principle......................... 52 6.3 The Heisenberg and Schrödinger pictures....................... 52 6.4 The Heisenberg equation of motion........................... 53 6.5 Conserved quantities................................... 53 6.6 Ehrenfest’s theorem................................... 54 6.7 Quantum numbers.................................... 55 7 Quantum mechanics 57 7.1 Infinite-dimensional Hilbert spaces........................... 58 7.2 Bases........................................... 58 Lecture CONTENTS 3 PX2132: Introductory Quantum Mechanics 7.3 The Fourier transform.................................. 60 7.4 Operators in the position basis............................. 61 7.5 Expectation values of operators............................. 61 7.6 Hermiticity of differential operators........................... 62 7.7 Basis-indepdendent TDSE................................ 63 7.8 The Postulates of Quantum Mechanics......................... 63 8 The quantum harmonic oscillator 65 8.1 Hermite polynomials................................... 66 8.2 Ladder operators..................................... 67 8.2.1 Commutation relations.............................. 68 8.2.2 Energy eigenstates and eigenvalues....................... 68 8.2.3 Normalization................................... 71 8.3 Second quantization................................... 72 9 The Schrödinger equation in three dimensions 73 9.1 TISE in three dimensions................................ 74 9.1.1 Cubic box..................................... 74 9.1.2 3D Harmonic oscillator............................. 76 9.2 Angular momentum................................... 77 9.2.1 Cartesian co-ordinates.............................. 77 9.2.2 Spherical polar co-ordinates........................... 78 9.2.3 Angular momentum ladder operators...................... 78 10 The hydrogen atom 80 10.1 Spherically-symmetric potentials............................ 81 10.1.1 Angular equation................................. 81 10.1.2 Radial equation.................................. 83 10.1.3 Solution and normalization........................... 84 10.2 The hydrogen atom.................................... 85 10.2.1 The Bohr model................................. 87 Lecture CONTENTS 4 PX2132: Introductory Quantum Mechanics Books All the information relevant to this course appears in a condensed form in the accompanying notes. The online videos provide further detail. There is no course textbook, but the notes provide references to the following books when helpful (all are freely available online): J. Binney and D. Skinner, The Physics of Quantum Mechanics [https://www-thphys.physics.ox.ac.uk/people/JamesBinney/QBhome.htm] P. A. M. Dirac, The Principles of Quantum Mechanics [archive.org/details/in.ernet.dli.2015.177580] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics [feynmanlectures.caltech.edu] While later editions of the first two books are available for a price, references should be assumed to be made to these free editions. The following books are available as eBooks for free through Cardiff University library and will also be referred to: D. J. Griffiths, Introduction to Quantum Mechanics (Cambridge University Press, 2nd edi- tion) S. Weinberg, Lectures on Quantum Mechanics (Cambridge University Press, 2nd edition, 2015). Other books which you may wish to consult, but which will not be referred to directly in the course: A. I. M. Rae and J. Napolitano, Quantum Physics (Routledge, 6th edition, 2015) ISBN 9781482299182 J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics (Cambridge University Press, 2nd edition, 2017) ISBN 978-1-108-42241-3 L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Volume 3 - Quantum Me- chanics: Non-Relativistic Theory (Pergamon Press, Third edition, 1977) ISBN 0080291406 S. Gasiorowicz, Quantum Physics (Wiley, 3rd edition, 2003) ISBN 978-0471057000 A. P. French and E. Taylor, An Introduction to Quantum Physics (W. W. Norton & Company, 1978) ISBN 0393091066 R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Par- ticles (Wiley and Sons, 2nd edition, 1985) ISBN 978-0471873730. Lecture CONTENTS 5 PX2132: Introductory Quantum Mechanics List of definitions The canonical commutation relation [x̂, p̂] = i~Î Dirac notation the notation |ψi for complex vectors. Also called bra-ket notation, with hφ| the ‘bra’, |ψi the ‘ket’, and hφ|ψi a bracket. Expectation value hÂi = hψ|Â|ψi. The mean value of an operator measured by a given state. First quantization a wave-like description of quantum objects: ψ (x). The Hamiltonian the energy operator (assumed time independent in this course). Ĥ = p̂2 /2m+ V̂ , or Ĥψ (x) = −~2 ψ 00 /2m + V (x) ψ. The Heisenberg picture the description of quantum states as time independent, and operators as time dependent. Dh iE The Heisenberg uncertainty principle σÂ σB̂ ≥ 1 2 Â, B̂ where σÂ denotes the standard deviation of operator Â. Hilbert space a linear vector space with an inner product and square-normalisable vectors R∞ ∗ Hermiticity Â = Â† where Â† = Â∗T. For differential operators: −∞ ϕ (x) Âψ (x) dx = R∞ ∗ −∞ Âϕ (x) ψ (x) dx. Ladder operators an operator which raises or lowers the quantum number of a state it acts on. Also called creation/annihilation operators or raising and lowering operators. Normalisation the prefactor on a wavefunction ensuring that the total probability to find the particle is one. The number operator in the harmonic oscillator, the operator whose eigenstates are the energy eigenstates and whose eigenvalues are the level of the state. Operators objects which act on states to give states (either the same state, or a different one). In fininte dimensional Hilbert spaces these are simply matrices acting on vectors to give vectors. In infinite dimensional Hilbert spaces these are differential operators acting on functions to give functions. Orthonormality orthogonal and normalised. If a set of states is orthonormal the inner product of any state with itself is 1 and the inner product between any two different states is zero. The probability density ρ integrated over a region of space, this gives the probability to find 2 the particle in that region. ρ (x, t) dx = |ψ (x, t)| dx is the probability to find the particle between x and x + dx at time t. The probability current density j the current density associated with a flow of probability: j (x, t) = i~ 2m {ψ∇ψ ∗ − ψ ∗ ∇ψ}. Lecture CONTENTS 6 PX2132: Introductory Quantum Mechanics The probability amplitude the complex number associated to each point in space by the wave- function ψ. Quantum numbers eigenvalues of operators which commute with the Hamiltonian; expectation values which do not change in time. The Schrodinger picture the description of quantum states as time dependent, and operators as time independent. Second quantisation a particle-like description of quantum objects in terms of ladder operators. Stationary states energy eigenstates. So called as their probability densities are time indepen- dent. Superposition summing solutions to the TDSE to get a new solution to the TDSE The time dependent schroedinger equation i~∂t |ψi = Ĥ|ψi, or in the position basis i~ψ̇ = Ĥψ. Abbreviated TDSE. The time independent schroedinger equation Ĥ|ψi = E|ψi, or in the position basis Ĥψ (x) = Eψ (x). Abbreviated TISE. The wavefunction ψ a function which assigns a complex number to each point in space. The modulus square is the probability density ρ. Lecture CONTENTS 7 PX2132: Introductory Quantum Mechanics 1 The Motivation for Quantum Mechanics Videos: V1.0: Introduction to the course V1.1: History of quantum mechanics V1.2: The Schrödinger equation V1.3: Plane waves V1.4: Amplitudes and probabilities V1.5: Two slit demo Topics: the time-dependent Schrodinger equation (TDSE) the time-independent Schrodinger equation (TISE) the wavefunction probability density probability current density general boundary conditions. For the exam you should be able to: understand and apply p = ~k, E = ~ω write down the TDSE and TISE state the assumptions going into the TDSE derive the TISE from the TDSE using separation of variables deduce the time dependence of a solution to the TISE state the Born rule state the meaning of the probability density and to calculate it for a given wavefunction derive the continuity equation and explain the meaning of local conservation of probability derive the probability current density and calculate it for a given wavefunction state the general boundary conditions on the wavefunction. Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 8 PX2132: Introductory Quantum Mechanics 1.1 The Schrödinger equation 1.1.1 The time-dependent Schrödinger equation (TDSE) The time-dependent Schrödinger equation (TDSE) is i~∂t ψ (x, t) = Ĥψ (x, t) (1) where ∂ψ (x, t) ∂t ψ (x, t) = (2) ∂t x and (·)x indicates that x is held constant. Here, ~ is the reduced Planck’s constant, x is position, t is time, and i is the imaginary unit. The function ψ (x, t) is called the wavefunction. It is difficult to overstate the importance of this equation, which won Erwin Schrödinger the Nobel Prize for Physics in 1933. It was the first solid mathematical description of particles in the microscopic world. Some points to note: Schrödinger did not derive it, he postulated it and checked its results match to reality. it is compatible with the de Broglie and Einstein relations, p = ~k and E = ~ω. it is linear, meaning that the solutions ψ obey linear wave superposition: any two solutions add to another solution. 1.1.2 The wavefunction ψ The solutions to the TDSE are complex wavefunctions ψ (x, t). These assign a complex number to every point in space and time; looked at the other way, if you plug in a value of x and t into ψ, then ψ gives you back a complex number. In essence, quantum mechanics simply involves solving the TDSE to find ψ (x, t) for a variety of boundary conditions, and interpreting the results. The reason for this is that ψ gives the maximum amount of information we believe it is possible to obtain about the corresponding physical system. That might seem surprising, but think of more familiar waves, such as water waves: these, too, are described by a wavefunction, which entirely specifies the form of the wave for all space and time. All this requires is a function which tells us two things at each position and time: 1. the amplitude of the wave (how high it is) 2. the phase of the wave (how far along in the cycle of going up and down it is). The same is true of ψ in quantum mechanics. In this case the equivalent to the amplitude of the wave is the square modulus of ψ, which relates to the probability to find a particle. The phase becomes meaningful when waves interfere, as with classical waves. Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 9 PX2132: Introductory Quantum Mechanics 1.1.3 Assumptions going into the TDSE The TDSE stated above is valid for particles which are: non-relativistic non-interacting massive (as in, have any non-zero mass). Generally we will assume the particle to be an electron. It was by solving the equation for the case of a single electron moving in the potential of a fixed proton, and showing that it agreed accurately with the behaviour of the hydrogen atom, that Schrodinger saw the validity of the equation. 1.1.4 The Hamiltonian operator The Hamiltonian operator Ĥ, which has dimensions of energy, is defined by the equation ~2 2 Ĥψ (x, t) =− ∇ + V (x) ψ (x, t) (3) 2m 2 ~ =− ∇2 ψ (x, t) + V (x) ψ (x, t) (4) 2m where m is the mass of the particle, V (x) is the potential confining the particle, and   ∂x   ∇ =  ∂y  (5)     ∂z is the gradient operator (I have used the simpler notation ∂x = (∂/∂x)y,z,t here). Operators: the idea of an ‘operator’ such as ∇ or Ĥ might seem mysterious. What does a derivative mean if there is no function on which it acts? The easy way to understand them is to just recall that they will always be acting on some function in the calculation: if you see ∇, just imagine ∇ψ. Some calculations can be simplified by only thinking about the operators, but you can think of this as a shorthand for familiar derivatives. The TDSE is a partial differential equation (PDE), meaning it is a differential equation written in terms of derivatives which hold some of the variables constant. The TDSE is second order in spatial derivatives and first order in time derivatives. Fortunately there is a trick which allows us to separate off the time dependence by simply solving an ordinary differential equation (ODE), familiar from first year. The trick is called separation of variables. Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 10 PX2132: Introductory Quantum Mechanics 1.1.5 Separation of variables Whenever you see the method of separation of variables it always follows the same form. Let’s look at it in the TDSE. First, write the wavefunction ψ (x, t) as the product of two functions, one of which is only a function of space φ (x), while the other is only a function of time T (t): ψ (x, t) = φ (x) T (t). (6) Not all PDEs are separable, but in this course we only care about the TDSE, which is. We will now substitute this new form into the TDSE, Eq.1: ∂ψ i~ = Ĥψ (7) ∂t x ∂ (φ (x) T (t)) i~ = Ĥ (φ (x) T (t)). (8) ∂t x But note that the derivative on the left holds all spatial components to be constant, while the Hamiltonian acts only on the spatial components and holds time constant. Therefore the ∂t sees φ (x) as a constant, and the ∇2 in the Hamiltonian (the only derivative) sees T (t) as a constant. We can therefore pull them out of the derivatives: ∂T (t) i~φ (x) = T (t) Ĥφ (x). (9) ∂t Since T is only a function of t, we can replace the partial derivative with a total derivative: dT (t) i~φ (x) = T (t) Ĥφ (x). (10) dt Dividing both sides by φT then leaves us with a left hand side which only contains functions of t and their derivatives, and a right hand side which contains only functions of position and their derivatives: ↓ 1 dT (t) 1 i~ = Ĥφ (x). (11) T (t) dt φ (x) Now, if you think about it, the only way both sides of the equation can hold is it they are both equal to the same constant. The reason is that the left hand side is independent of space, so could only equal something varying in time, while the right hand side is independent of time, so could only equal something varying in space. The only function which satisfies both is a constant. Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 11 PX2132: Introductory Quantum Mechanics Let us suggestively call the constant E. We can assume E is real for now; we will prove this later. This gives two equations. The first, from the right hand side, is the time-independent Schrödinger equation (TISE): 1 Ĥφ (x) = E φ (x) ↓ Ĥφ (x) = Eφ (x). (12) The second gives the time evolution of the wavefunction: 1 dT (t) i~ =E T (t) dt ↓ T (t) = exp (−iEt/~) T (0) ↓ ψ (x, t) = ψ (x, 0) exp (−iEt/~). (13) Notation: throughout this course we will use the symbol ψ (x, t) to refer to solutions to the TDSE, and φ (x) to refer to solutions to the TISE. This last expression is key, because it means that if we can solve the TISE we get the solutions to the TDSE for free. If you think about it that’s quite remarkable – if you can specify the wavefunction perfectly at an instant in time, you know its form for all past and future times! The statement is true in the absence of measurement, which is far more mysterious. Actually the predictive power of the TDSE is not so mysterious: recall your classical sound waves, governed by the (classical) wave equation. The same is true there. 1.2 How to Solve the Schrödinger Equation Quantum mechanics has survived because it gives phenomenally accurate predictions. The process of obtaining predictions in quantum mechanics is the process of solving the TDSE. This is always done in the same way. Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 12 PX2132: Introductory Quantum Mechanics First, specify the problem by writing down the potential V (x). It always helps to draw the potential. Second, solve the TISE, Eq. 12. In 1D this is a 2nd order ODE in position x; in 2D or 3D it is a PDE (there are now y and z to deal with), but many cases of interest remain solvable as we will see. As you may recall from first year maths, solving a 2nd order ODE always takes the form of – (i) recognising the general solution – (ii) applying the boundary conditions. Third, armed with this information we can straightforwardly add the time dependence to the solutions we have found using Eq 13. This gives the desired set of solutions to the TDSE (what we wanted). Finally, and very importantly: Quantum mechanics is linear. This means that adding any two solutions to the TDSE gives another solution to the TDSE. Often in these notes we will simply solve the TISE and will not bother getting the time dependence back out, as the process is always the same. But remember: it’s the solutions to the TDSE we really want. I heard this put nicely once: Schrödginer won the Nobel prize for writing down the TDSE, not the TISE. The TISE is just a means to solving the TDSE. This is easy to forget since in practice we spend all our time solving the TISE, as it’s the hard part of the problem. 1.3 General boundary conditions The following boundary conditions apply to all the cases of physical interest in the TISE: (i) φ (x) is continuous (ii) ∇φ (x) is continuous except possibly at infinite discontinuities in V (x) (14) (iii) φ = 0 in continuous regions where V = ∞. The first two are simply properties of differential equations, and follow from the fact that the TISE is second-order in spatial derivatives, and we never consider potentials which are too pathological (we can have infinite potentials and discontinuous potentials, and even derivatives of discontinuities, but nothing worse). The third can be thought of physically as having the particle avoid regions of infinite potential, which makes physical sense (it is also born out by the mathematics). Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 13 PX2132: Introductory Quantum Mechanics 1.4 Probability density The meaning of the wavefunction ψ is debated. What can be said for certain is the Born rule: 2 The Born rule: |ψ (x, t)| dx gives the probability to find the particle between x and x + dx at time t. This rule wasn’t obvious to anyone, but many experimental tests have now been carried out to confirm it. This is one of the most directly testable results of quantum mechanics: the probability to find a particle between points a and b (along a line) is given by Z b 2 Prob(a < x < b) = |ψ| dx. (15) a You can think of dx as being like a little line segment. Thinking about the units, the quantity 2 ρ (x, t) = |ψ (x, t)| (16) must therefore be a probability density: a thing which, when integrated over space, gives a proba- bility. Since the particle must exist somewhere in the universe we a general rule: Z ∞ ρ (x, t) dx = 1. (17) −∞ The overall probability to find the particle somewhere is always 1, no matter what happens. The technical way to say this is that probability is globally conserved. 1.5 Probability current density Consider how the probability density ρ (x, t) changes with respect to time (it’s no harder to consider three dimensions, so we will for generality): 2 ∂t ρ (x, t) = ∂t |ψ| = ψ ∗ ∂t ψ + ψ∂t ψ ∗ 1 = {ψ ∗ (i~∂t ψ) − ψ (−i~∂t ψ ∗ )} i~ ↓ TDSE i~ ∗ 2 = ψ ∇ ψ − ψ∇2 ψ ∗ 2m i~ = ∇ · {ψ ∗ ∇ψ − ψ∇ψ ∗ }. 2m This gives us a continuity equation for conservation of probability: ∂t ρ (x, t) = −∇ · j (x, t) (18) Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 14 PX2132: Introductory Quantum Mechanics where i~ j (x, t) = {ψ∇ψ ∗ − ψ ∗ ∇ψ} (19) 2m is the probability current density. The physical meaning is that probability is locally conserved. This is a stronger statement than global conservation. It says that if the probability is going to decrease in one region of space, it must flow out of that region, like a fluid. If a quantity were globally but not locally conserved, it could decrease in one region and increase in a disconnected region simultaneously. For example, when you shuffle cards, the number of red cards in the pack is conserved globally but not locally. Lecture 1 THE MOTIVATION FOR QUANTUM MECHANICS 15 PX2132: Introductory Quantum Mechanics 2 Scattering and tunnelling Videos: V2.1a–V2.1d: Scattering from a potential step V2.2: Quantum tunnelling V3.3: Evanescent waves demo Topics: plane waves recovering p = ~k and E = ~ω from the Schrödinger equation scattering from a potential step tunnelling and barrier penetration scanning tunnelling microscopes. For the exam you should be able to: write down the form of a plane wave use this form to show the Schrödinger equation is compatible with p = ~k and E = ~ω state the forms of the TISE in regions of constant potential find the transmission and reflection amplitudes for scattering from a potential step find the probability current densities for scattering from a potential step find the probabilities of transmission and reflection from a potential step explain the steps necessary to solve scattering from a potential barrier of finite width explain the physical significance of quantum tunnelling explain the relevance to scanning tunnelling spectroscopy and microscopy For the exam you will not be required to: rote learn any solutions solve explicitly for the amplitudes associated with the finite-width barrier Lecture 2 SCATTERING AND TUNNELLING 16 PX2132: Introductory Quantum Mechanics 2.1 TISE: solutions in regions of constant potential In regions of constant potential there are three possible forms of solution, all of which have coun- terparts in classical waves: travelling waves (plane waves):φ (x) = a exp (ikx) + b exp (−ikx) (20) standing waves:φ (x) = a cos (kx) + b sin (kx) (21) evanescent waves:φ (x) = a exp (κx) + b exp (−κx). (22) The coefficients a and b can be complex in general. The forms above assume k and κ are real. In fact any one of the three options is already completely general if we allow complex wavevectors. Nevertheless, it is often simpler to choose the appropriate form of solution using physical intuition. Standing waves are relevant if leftgoing and rightgoing travelling waves appear in the region with equal amplitude (useful for bound states); evanescent waves are relevant if the particle’s energy is less than the potential in the region. 2.2 Plane waves A trivial example is given by the potential V (x) = 0. Here we find the general solutions φ (x) = a± exp (±ikx) T (t) = T (0) exp (−iEt/~) (23) for real k. The time dependent wave function is ψ (x, t) = φ (x) T (t) = a± exp (±ikx − iEt/~) (24) for a possibly complex a±. These correspond to plane waves propagating in the ±x directions. Comparing to the general form of a plane wave solution ψ (x, t) = a± exp (±ikx − iωt) , (25) which should hopefully be familiar from other courses, we see that E = ~ω. (26) This is Einstein’s relation relating the energy of a particle to its angular frequency. Substituting Eq 25 into the TISE, Eq. 12, we find that Lecture 2 SCATTERING AND TUNNELLING 17 PX2132: Introductory Quantum Mechanics Figure 1: The potential step defined in Eq. 30, with schematic solutions indicated. ~2 k 2 E=. (27) 2m Equating this to the non-relativistic expression for the kinetic energy p2 E= (28) 2m we find the de Broglie relation p = ~k. (29) 2.3 Scattering from a potential step Consider the potential   0, x < 0 (region I)  V (x) = (30)  V0 , x ≥ 0 (region II).  This is shown schematically in Fig. 30. The picture draws an intuitive analogy with graviational potential. But the particle really lives along a 1D line, and the potential would generally be electrostatic. Assume a particle is incident on the step from the left. To find the wavefunction which solves the TISE we can solve the TISE in regions I and II and match them using the the general boundary conditions of Eq. 14. From the solution of the TISE we can always obtain the full solution to the TDSE straightforwardly using Eq. 13. We will return to this later. For now we will focus on solving the TISE. 2.3.1 Region I For x < 0 we have the general solution φI (x) = aI exp (ikI x) + bI exp (−ikI x) ~2 kI2 E=. (31) 2m where kI are real. There is no reason to assume the leftgoing and rightgoing amplitudes are the same, so standing waves would not be appropriate. The energy of the particle is greater than that of the (zero) potential in this region, so we expect wave-like solutions rather than evanescent solutions. The expression for E in terms of kI is again found by substituting the form of φ into Lecture 2 SCATTERING AND TUNNELLING 18 PX2132: Introductory Quantum Mechanics the TISE. Here aI corresponds to the rightmoving ingoing wave, and bI the leftmoving reflected wave. This wavefunction is unphysical as it cannot be normalised; a more physical solution would involve summing up plane waves to form a wave packet of finite extent. Without loss of generality we can choose to set aI = 1 and bI = r, the reflection amplitude: φI (x) = exp (ikI x) + r exp (−ikI x). (32) 2.3.2 Region II: E > V0 Assuming E > V0 we have the general solution φII (x, t) = aII exp (ikII x) + bII exp (−ikII x) ~2 kII 2 E − V0 =. (33) 2m Since the energy E is the same in Eqs 31 and 33 we have 2mV0 kI2 = kII 2 +. (34) ~2 Physically we know that bII = 0, because this term corresponds to a left-going wave. But the wave entered from the left, so must be purely right-going. Rename aII = t the transmission amplitude: φII (x) = t exp (ikII x). (35) Now we can use the general boundary conditions (i) and (ii) which are always true (Eq 14). Boundary condition (i) gives: φI (0) = φII (0) ↓ 1 + r = t. (36) Boundary condition (ii): φ0I (0) = φ0II (0) ↓ kI (1 − r) = kII t. (37) Lecture 2 SCATTERING AND TUNNELLING 19 PX2132: Introductory Quantum Mechanics Combining with Eq. 36 we have: kI − kII r= (38) kI + kII 2kI t=. (39) kI + kII These are the probability amplitudes for reflection and transmission. Their meaning is obscure in exactly the same way that the meaning of ψ is obscure; but they can be used to establish the probability for reflection and transmission, which we will do shortly. First let’s consider the case E < V0. 2.3.3 Region II: E < V0 Assuming E > V0 we instead have the solution φII (x) = aII exp (κx) + bII exp (−κx) ~2 κ2 E − V0 = −. (40) 2m We know on physical grounds that aII = 0. We can rename bII = t. φII (x) = t exp (−κx). (41) Boundary condition (i): φI (0) = φII (0) ↓ 1+r =t (42) as before (Eq. 36). From Eqs. 31, 40 2mV0 kI2 = −κ2 +. (43) ~2 Boundary condition (ii): Lecture 2 SCATTERING AND TUNNELLING 20 PX2132: Introductory Quantum Mechanics φ0I (0) = φ0II (0) ↓ ikI (1 − r) = −κt. (44) Therefore combining with Eq. 42 we have: kI − iκ r= (45) kI + iκ 2kI t=. (46) kI + iκ Note that the same result can be achieved more simply by substituting kII → iκ into the results for E > V0. 2.3.4 Probability fluxes Now to extract some physically measurable quantity from the results. The probability that the particle is reflected by the step, R, is given by the ratio of reflected probability current density jR to incident probability current density jin : jR R= (47) jin where the probability current density is defined in Eq. 19. Similarly the probability that the particle is transmitted by the step, T , is given by the ratio of transmitted probability current density jT to incident probability current density jin : jT T =. (48) jin First consider the case E > V0. Using Eq 13 to establish the time dependence of the wavefunctions we have ψin (x, t0 ) = exp (−iEt0 /~) exp (ikI x) (49) ψR (x, t0 ) = r exp (−iEt0 /~) exp (−ikI x) (50) ψT (x, t0 ) = t exp (−iEt0 /~) exp (−ikII x) (51) where t0 has been used to indicate time in order to avoid confusion with the transmission probability amplitude t. Using the expression for the probability current density in terms of the wavefunction (Eq 19) gives Lecture 2 SCATTERING AND TUNNELLING 21 PX2132: Introductory Quantum Mechanics ~kI jin = (52) m ~ (−kI ) 2 jR = |r| (53) m ~k 2 jT = II |t| (54) m where the negative sign in jR is because j is a vector quantity. The probability of reflection and transmission is then 2 R = |r| (55) kII 2 T = |t|. (56) kI Substituting the forms of r and t found above gives 2 kI − kII R= (57) kI + kII 2 kII 2kI T =. (58) kI kI + kII It is straightforward to verify that R + T = 1. (59) This is just a statement of the conservation of probability. For E < V0 we have ψin (x, t0 ) = exp (−iEt0 /~) exp (ikI x) (60) ψR (x, t0 ) = r exp (−iEt0 /~) exp (−ikI x) (61) ψT (x, t0 ) = t exp (−iEt0 /~) exp (−κx) (62) and so ~kI jin = (63) m ~ (−kI ) 2 jR = |r| (64) m jT = 0. (65) The probability of reflection and transmission is now Lecture 2 SCATTERING AND TUNNELLING 22 PX2132: Introductory Quantum Mechanics Figure 2: The potential barrier of Eq. 68. Schematic solutions; those in region II assume E < V0. R=1 (66) T = 0. (67) The particle is reflected with certainty. It nevertheless has an amplitude to be detected within the barrier. This is a manifestation of some of the magic of quantum mechanics. Even though the particle is certainly reflected from the barrier, if you look for it in the barrier you may find it. For a measurement to locate the particle in the barrier, it would need to provide enough energy for that event to occur. The additional energy would have to be provided by the measurement itself. You can think of it as the measurement device changing the potential so that the potential step has a finite width, beyond which E < V0 again. We will consider this situation next. 2.4 Quantum tunnelling Now consider the potential     0, x < 0 (region I)   V (x) = V0 , 0 ≤ x < L (region II) (68)    x ≥ L (region III).   0, This is shown in Fig. 2. We can no longer neglect the increasing solution in region II. While we therefore have two extra unknowns, we also have two extra boundary conditions, giving four in total: (i) φI (0) = φII (0) (69) (ii) φ0I (0) = φ0II (0) (70) (iii) φII (L) = φIII (L) (71) (iv) φ0II (L) = φ0III (L). (72) We know that kI = kIII since the potentials are the same in these regions: kI = kIII , k (73) and substituting into the TISE gives Lecture 2 SCATTERING AND TUNNELLING 23 PX2132: Introductory Quantum Mechanics √ 2mE k=. (74) ~ As before, we must treat the cases E > V0 and E < V0 separately. 2.4.1 E > V0 The wavefunctions in the various regions are: φI (x) = exp (ikx) + r exp (−ikx) (75) φII (x) = a exp (ik 0 x) + b exp (−ik 0 x) (76) φIII (x) = t exp (ikx). (77) where the wavevector in region II is defined to be k 0 : kII , k 0 (78) p 2m (E − V0 ) k0 =. (79) ~ Applying the boundary conditions we have (i) 1+r =a+b (80) 0 k (ii) 1−r = (a − b) (81) k (iii) a exp (ik 0 L) + b exp (−ik 0 L) = t exp (ikL) (82) 0 k (iv) (a exp (ik 0 L) − b exp (−ik 0 L)) = t exp (ikL). (83) k These are four coupled linear equations in four unknowns, and can be solved by standard methods. There is no especially elegant way to do so. If you work through the problem, which is beyond the scope of this course (but do-able), you can solve for the probability of transmission: 4E (E − V0 ) T = p . (84) 4E (E − V0 ) + V02 sin2 L 2m (E − V0 )/~ Note that when k 0 L = nπ for integer n, there is perfect ‘resonant transmission’, T = 1, R = 0. The particle passes through the barrier as if it weren’t there. Lecture 2 SCATTERING AND TUNNELLING 24 PX2132: Introductory Quantum Mechanics 2.4.2 E < V0 The wavefunction regions I, III are unchanged, and that in region II is: φII (x) = a exp (κx) + b exp (−κx). (85) We can simply take the results of the E > V0 case and substitute k 0 → −iκ. Substituting the expressions for k and κ: 4E (V0 − E) T = p . (86) 4E (V0 − E) + V02 sinh2 L 2m (V0 − E)/~ Resonant transmission is no longer possible. But the result is nevertheless remarkable. The particle has a probability to be found on the opposite side of the barrier to where it started, even though the potential of the barrier is larger than the energy of the particle. This is quantum tunnelling. Lecture 2 SCATTERING AND TUNNELLING 25 PX2132: Introductory Quantum Mechanics 3 Bound states Videos: V3.1: The infinite potential well V3.2: Normalisation V3.3: The finite potential well Topics: The infinite potential well (particle in a box) Finding the energy eigenvalues and eigenfunctions the Born rule normalisation of wavefunctions the finite potential well For the exam you should be able to: solve for the energy eigenstates and eigenvalues of the infinite potential well (particle in a box) explain the physical relevance of normalisation normalise a given wavefunction justify the forms of the wavefunctions solving the TISE for the finite potential well explain the steps involved in solving the TISE in the finite potential well prove that there is at least one bound state in any finite potential well For the exam you will not be required to: provide a full solution for the finite potential well Lecture 3 BOUND STATES 26 PX2132: Introductory Quantum Mechanics Figure 3: The infinite potential well. Snapshots of the real parts of the first few energy eigenfunc- tions are shown, offset vertically for clarity (see video V3.1). 3.1 The 1D infinite potential well This is the classic problem in quantum mechanics, as it can be solved exactly and straightforwardly. A particle is constrained to a 1D line, as before; now the potential is either infinity (in which case the particle can’t exist in that region) or it is zero, in which case the TISE is easy to solve. Consider the following potential:   0,  0 0 (171) (i.e. the square modulus of a vector is always strictly greater than zero) we see that M = M † ⇒ λ = λ∗ (172) i.e. Hermitian operators have real eigenvalues The normalised eigenvectors of Hermitian matrices are orthonormal. Begin again with Eq. 168. Consider this object: hvm |M − M † |vn i = (λn − λm ) hvm |vn i (173) where the equality follows from acting M to the right and M † to the left as before. Assuming no two eigenvalues are degenerate, i.e. λn 6= λm ∀m 6= n, we have that M = M † ⇒ hvm |vn i = 0. (174) Furthermore, assuming we always normalise our eigenvectors correctly, hvn |vn i = 1 (175) and so we have the stronger condition hvn |vm i = δnm (176) with δnm the Kronecker delta. Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 44 PX2132: Introductory Quantum Mechanics In fact the eigenvectors of a Hermitian matrix form a complete orthonormal basis for the vector space (Hilbert space) in which they live. This means that we can use them to resolve the identity: X I= |vn ihvn |. (177) n Acting on this expression with M shows that any Hermitian operator can be written as the sum of its eigenvalues multiplied by outer products formed from their respective eigenvectors: X M= λn |vn ihvn |. (178) n In fact, you can look at the resolution of the identity as a special case of this, since the identity matrix is Hermitian with eigenvalue 1 for every vector. Therefore any complete orthonormal basis can be used to construct the identity. 5.3 Spin-1/2 Reference: Feynman III Chapter 5 5.3.1 Mathematical structure A simple example of matrices in action in quantum mechanics is provided by spin-1/2 particles (of which electrons are an example). Spin is intrinsic angular momentum. Classically you might think of it as analogous to the angular momentum of the Earth spinning about its axis, but it is inherently quantum in nature and has no good classical analogue. In the Stern Gerlach experiment a beam of spin-1/2 silver atoms is directed through a magnetic field gradient (directed along z). This accelerates the atoms according to the z-projection of their spins. Silver atoms were used originally as these have spin but are charge neutral. Electrons would receive an additional unhelpful redirection owing to their charges. Classically we would expect a continuous range of deflections. Measuring the Earth’s intrinsic angular momentum along a continuous range of directions will return a continuous range of val- ues, with a maximum when measuring parallel to the axis of rotation and zero when measuring perpendicular to this axis. The silver atoms were instead found to deflect in one of two directions, a clear demonstration of quantization. For spin-n particles there are 2n + 1 directions. Some key observations are these: measurement of spin along any chosen direction yields either +~/2 or −~/2 subsequent measurements along the same direction, without other measurements in between, will consistently return the same result Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 45 PX2132: Introductory Quantum Mechanics if the spin is known along z, a measurement along any perpendicular direction is completely ambiguous, with 50% probability for each of ±~/2 therefore, even though repeated measurements along z yield the same result, if a measurement along x is performed, a subsequent re-measurement of z will give a 50% probability of either result ±~/2. This sequence of observations gets to the heart of quantum weirdness. Quantum mechanics is not magic for the sake of magic; it is the simplest explanation we know of which can explain the behaviour of the world on the smallest scales. The observations above suggest the following mathematical structure. The state of the spin- 1/2 particle should be represented by vectors with two eigenvalues, ±~/2. Since there are two eigenvalues there should be two eigenvectors which span a two-dimensional complex vector space. The observable quantities associated with spin measurements should be 2 × 2 Hermitian matrices with these eigenvalues. Defining Ŝi to be the operator corresponding to the observable spin along direction i, we have: ~ Ŝi | ↑ii = + | ↑i i, i ∈ {x, y, z} (179) 2 ~ Ŝi | ↓ii = − | ↓i i. (180) 2 There should be three such matrices, one for each perpendicular direction, and they should not commute. Their lack of commutation should be such that an eigenvector of one operator should have equal amplitudes to be either eigenvector of either other operator. For example: 1 | ↑x i = √ exp (iα) (| ↑z i + exp (iβ) | ↓z i) (181) 2 where α is a real number corresponding to an unmeasurable global phase, and β is a real number corresponding to an unspecified relative phase. Similar relations must hold between the other eigenvectors. One consistent choice is as follows: ~ Ŝi = σi (182) 2 with Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 46 PX2132: Introductory Quantum Mechanics   0 1 σx =   (183) 1 0   0 −i σy =   (184) i 0   1 0 σz =  . (185) 0 −1 These are called the Pauli matrices. For this choice we have   1 | ↑z i =   (186) 0   0 | ↓z i =   (187) 1 and, for example,   1  1  1 | ↑x i = √ = √ (| ↑z i + | ↓z i). (188) 2 1 2 The different matrices do not commute, as required: h i Ŝi , Ŝj = i~ijk Ŝk (189) where Einstein summation notation has been assumed (i.e. there is an implicit sum over index k) and ijk is the Levi-Civita symbol defined by     0, any of i, j, k equal   ijk = 1, ijk = 123 or cyclic permutations (190)     −1, ijk = 321 or cyclic permutations.  5.3.2 Measurements To work out what happens when a state is prepared with a definite state of spin in one direction but is then measured in a different direction, we have to decompose the vector into the relevant basis. For example, say a state is prepared with spin up along x:   1 1 |ψi = | ↑x i = √  . (191) 2 1 Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 47 PX2132: Introductory Quantum Mechanics This state is then measured in the z direction. To work out the possible outcomes and their relative amplitudes we should write the state in the z-basis:   1 1 | ↑x i = √   = √1 (| ↑z i + | ↓z i). (192) 2 1 2 The probability for measuring | ↑z i is given by the square magnitude of the coefficient of | ↑z i: 1/2. Mathematically, the probability amplitude for finding the state | ↑z i upon performing a measurement in the z direction for a state | ↑x i is h↑z | ↑x i. (193) In this case, 1 h↑z | ↑x i = h↑z | · √ (| ↑z i + | ↓z i) (194) 2 1 = √ (h↑z | ↑z i + h↑z | ↓z i) (195) 2 1 = √ (1 + 0) (196) 2 and the corresponding probability for measuring | ↑z i in a z-measurement of state | ↑x i is 2 |h↑z | ↑x i| = 1/2. (197) In general we can say that: the amplitude for measuring state |ϕi when a state is prepared as |ψi is hϕ|ψi (198) and the corresponding probability is 2 |hϕ|ψi|. (199) We can also consider repeated measurements by passing the particle along multiple Stern Gerlach apparatuses with different orientations. By blocking off one of the two output paths of each apparatus we can select spins with a chosen orientation along a chosen direction. To find the amplitude for a state initially prepared spin-up-in-z to pass a filter for spin-down-in-y then spin-up-in-x, and to finally be measured spin-down-in-z, we need to evaluate Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 48 PX2132: Introductory Quantum Mechanics (amplitude for ↓z given initially ↑x ) · (amplitude for ↑x given initially ↓y ) · (amplitude for ↓y given initially ↑z ) h↓z | ↑x i · h↑x | ↓y i · h↓y | ↑z i (200) and so on. Each condition is required to be true independently so we multiply the probability amplitudes. Probability amplitudes in quantum mechanics play the role of probabilities in classical probability theory: it is the amplitudes which are multiplied or added rather than the probabilities. 5.3.3 The quantum eraser Consider a state initially spin up along z which is then passed through an x-oriented Stern Gerlach apparatus, but the two possible beams are recombined before measuring along z again. Then we must add the amplitudes for the two possibilities in the intermediate x state, because both are accepted: h↑z | ↑x i · h↑x | ↑z i + h↑z | ↓x i · h↓x | ↑z i. (201) But if we accept both intermediate states we haven’t made an intermediate measurement, so we don’t expect the state to be changed and the overall amplitude should be 1 (the amplitude for a state prepared spin-up-in-z to be measured spin-up-in-z). This is born out mathematically, as we can rewrite the expression as: h↑z | (| ↑x ih↑x | + | ↓x ih↓x |) | ↑z i (202) =h↑z |I| ↑z i (203) =h↑z | ↑z i = 1. (204) The second line follows because | ↑x i and | ↓x i together form a complete orthonormal basis for the set of 2D complex vectors, which must be true because they are the full set of non-degenerate eigenvectors of a Hermitian matrix Ŝx. You can also check this explicitly:     1  1  1  1  | ↑x i = √ , | ↓x i = √ (205) 2 1 2 −1 Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 49 PX2132: Introductory Quantum Mechanics     1 1 1 | ↑x ih↑x | + | ↓x ih↓x | =  (1, 1) + 1   (1, −1) (206) 2 1 2 −1     1 1 1 1 −1 =  + 1  (207) 2 1 1 2 −1 1   1 0 = . (208) 0 1 While a simple mathematical result, the physical interpretation is quite profound. Preparing a state of definite spin-up-in-z we pass the particle through an intermediate x-oriented Stern Gerlach apparatus which measures the spin in the x-direction. The two possible paths followed by the particle are separated macroscopically; you could put your finger in the gap between them. A measurement along x would randomise the result of a measurement along z. But because we erase the information about the x-measurement before measuring along z (by recombining the possible beams), the initial z information remains intact, even though a classical particle would take one beam or the other. This is an example of a quantum eraser experiment. It shows that there is a fundamental difference between a quantum superposition with two possible outcomes, versus a classical probability with two known outcomes (such as tossing a coin). Exactly how the quantum situation is explained falls to a matter of interpretation. 5.3.4 Expectation values The expected value of a given observable for a given state is simply given by the expectation value of the corresponding operator in that state. For example, the expected value of the spin in the i direction, of state |ψi, is hŜi i = hψ|Ŝi |ψi. (209) If we prepare a state of spin up in z, the expectation value of the Ŝz operator is ~ ~ h↑z |Ŝz | ↑z i = h↑z | | ↑z i = (210) 2 2 as expected since it is an eigenstate. We could have found this using matrices:    ~ 1 0 1  = ~. h↑z |Ŝz | ↑z i = (1, 0)  2 0 −1 0 2 On the other hand, Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 50 PX2132: Introductory Quantum Mechanics    ~ 0 1 1 h↑z |Ŝx | ↑z i = (1, 0)   = 0. (211) 2 1 0 0 This makes sense as a measurement of the spin in the x-direction of an eigenstate of spin in the z-direction has an equal probability of coming out +~/2 or −~/2. Lecture 5 FINITE-DIMENSIONAL HILBERT SPACES 51 PX2132: Introductory Quantum Mechanics 6 Matrix Mechanics Videos: V6.1: Operators and observables V6.2: The Heisenberg uncertainty principle V6.3: The Heisenberg picture V6.4: Conserved quantities Topics: p̂ and x̂ operators canonical commutation relations complete sets of states quantum numbers the Heisenberg uncertainty principle the Heisenberg and Schrodinger pictures the Heisenberg equations of motion Ehrenfest’s theorem. For the exam you should be able to: state the canonical commutation relations find the commutators of given operators explain the significance of operators commuting state the Heisenberg uncertainty principle show that given wavefunctions obey the uncertainty principle explain what is meant by the Heisenberg and Schrodinger pictures deduce the Heisenberg equation of motion state Ehrenfest’s theorem and explain its physical significance discuss the correspondence principle deduce Ehrenfest’s theorem from the Heisenberg equation of motion state the meaning of a conserved quantity Lecture 6 MATRIX MECHANICS 52 PX2132: Introductory Quantum Mechanics show that the observable quantity associated to a given operator is conserved explain the meaning of quantum numbers. For the exam you will not be required to: derive the Heisenberg uncertainty principle. Lecture 6 MATRIX MECHANICS 53 PX2132: Introductory Quantum Mechanics 6.1 Operators and observables Observable quantities are represented in quantum mechanics by Hermitian operators. Operators are denoted with hats Ô. For finite-dimensional Hilbert spaces operators are just Hermitian ma- trices. When dealing with wavefunctions we are dealing with infinite-dimensional Hilbert spaces (Leture 7). In this case operators are represented by differential operators such as −i~∇. This mathematical structure was chosen by Heisenberg based on experimental observations. First, real eigenvalues are required because we measure real numbers in experiment, and Hermitian operators have real eigenvalues. Second, different Hermitian matrices need not commute; if two matrices commute it is possible to find simultaneous eigenvectors for them, and the observables corresponding to both operators can be known simultaneously. But this need not be the case in quantum mechanics. For example, we have the position operator x̂. This has an infinite number of orthonormal eigenvectors |xi. The eigenvalues x are the possible positions of the particle: x̂|xi = x|xi. (212) We have the momentum operator p̂ with eigenvectors |pi: p̂|pi = p|pi. (213) The operators x̂ and p̂ do not commute. In fact we have the canonical commutation relation [x̂, p̂] = i~Î (214) where Î is the identity operator which has eigenvalue 1 for any state. When dealing with matrices it is the identity matrix. Because these operators do not commute, no state is able to simultaneously be an eigenstate of both. The canonical commutation relation encodes the experimental observation that it is not possible to have simultaneous knowledge of a particle’s position and momentum. The energy operator is already familiar. It is the Hamiltonian appearing in the Schrödinger equa- tion: Ĥ|ni = En |ni. (215) The energy receives contributions from the kinetic T̂ and potential V̂ terms. Being observable, each gets its own operator: p̂2 Ĥ = T̂ + V̂ = + V̂. (216) 2m In the special case of a free particle we can drop the potential term. In this case we have that Lecture 6 MATRIX MECHANICS 54 PX2132: Introductory Quantum Mechanics p̂2 Ĥ = (217) 2m and in this case the Hamiltonian commutes with the momentum operator: h i free particle: Ĥ, p̂ = 0. (218) When two operators commute it is possible to find simultaneous eigenvectors for them. Dirac notation is very convenient for such cases, allowing us to write things such as Ĥ|p, ni = En |p, ni (219) p̂|p, ni = p|p, ni. (220) 6.2 The Heisenberg Uncertainty Principle Proof: Griffiths 3.5.1 Define the uncertainty σÂ in an observable Â as the standard deviation: q σÂ = hÂ2 i − hÂi2. (221) For any two operators Â and B̂ we have the (generalized) Heisenberg Uncertainty Principle: 1 Dh iE σÂ σB̂ ≥ Â, B̂. (222) 2 For example, from Eq. 214 we have the uncertainty relation between x̂ and p̂: ~ σx σp ≥. (223) 2 Again, the maths is clear and undisputed. The physical interpretation comes down to your personal preference, until an experiment is deduced to distinguish the possible options. 6.3 The Heisenberg and Schrödinger pictures Until now we have been working in the Schrödinger picture: time-independent operators ÂS (S for Schrödinger) and time-dependent states |ψS (t)i, where |ψS (t)i = exp −iĤt/~ |ψS (0)i. (224) Here the state |ψi has not been assumed to be an energy eigenstate; if it is, the expression reduces to the usual form Lecture 6 MATRIX MECHANICS 55 PX2132: Introductory Quantum Mechanics |ψS (t)i = exp (−iEt/~) |ψS (0)i. (225) The Heisenberg picture takes the opposite approach: time-independent states |ψH i (H for Heisenberg) are acted on by time-dependent operators ÂH (t), where ÂH (t) = exp iĤt/~ ÂS exp −iĤt/~. (226) While Schrödinger operators can have their own explicit time dependence, for example if the potential is changing with time, we will not consider such cases in this course. The Schrödinger and Heisenberg pictures are completely equivalent and make all the same predictions. They are equivalent to active and passive transformations: operators map between states |ψi in the Hilbert space, and the question of time dependence is simply whether the states or operators are changing. In particular, amplitudes are the same in both pictures: hϕS (t) |ÂS |ψS (t)i = hϕS (0) | exp iĤt/~ ÂS exp −iĤt/~ |ψS (0)i = hϕH | exp iĤt/~ ÂS exp −iĤt/~ |ψH i = hϕH |ÂH (t) |ψH i. (227) Here we made the arbitrary choice |ψH i = |ψS (0)i; (228) other options are just changed by an unmeasurable global phase. 6.4 The Heisenberg equation of motion Taking the derivative of Eq. 226 with respect to time gives the Heisenberg equation of motion dÂH (t) h i i~ = ÂH (t) , Ĥ. (229) dt This equation fulfils the role of the TDSE in the Heisenberg picture. 6.5 Conserved quantities The observable quantity A associated with operator Â is conserved if and only if dhψ|Â|ψi A is conserved iff = 0 ∀|ψi. (230) dt Lecture 6 MATRIX MECHANICS 56 PX2132: Introductory Quantum Mechanics Note that this statement is independent of picture. It follows that h i A is conserved iff Â, Ĥ = 0. (231) Take the trivial but important example of Â = Î, the identity operator: h i Î, Ĥ = 0 (232) dhψ|ψi ∴ =0 (233) dt the conservation of probability. This was used in Eq. 19 to define the probability current. 6.6 Ehrenfest’s theorem In the Heisenberg picture states |ψH i are time independent, so sandwiching Eq. 229 between states gives dÂH (t) h i i~hψH | |ψH i = hψH | ÂH (t) , Ĥ |ψH i dt ↓ dhψH |ÂH (t) |ψH i h i i~ = hψH | ÂH (t) , Ĥ |ψH i. dt Noting that both sides of the expression are again independent of picture, we have Ehrenfest’s theorem: dhÂi h i i~ = h Â, Ĥ i. (234) dt This is key in demonstrating that quantum mechanics matches classical predictions on average, meaning that classical quantities relate to expectation values. In particular, we have dhx̂i m = hp̂i (235) dt which shows that the classical equation p = mv is obeyed by quantum systems when classical p and x are replaced by the expectation values of their quantum operators, and dhp̂i = −h∇V (x̂)i. (236) dt This resembles Newton’s second law, but not quite: the expectation value is taken over all h∇V (x̂)i, while the expression would need to read ∇V (hx̂i) for Newton’s second law to be obeyed by the Lecture 6 MATRIX MECHANICS 57 PX2132: Introductory Quantum Mechanics expectation values of position and momentum. Nevertheless, Ehrenfest’s theorem lends weight to the ‘correspondence principle’, that classical mechanics is returned in the limit of large quantum q 2 numbers: if the standard deviation hx̂2 i − hx̂i around the mean hx̂i is small, i.e. the particle is sharply localised, the approximation h∇V (x̂)i ≈ ∇V (hx̂i) is better, and classical mechanics is approximately returned by expectations values. Since x̂ and Ĥ do not commute, a wavepacket well- localised in x̂ necessarily requires more contribution from higher energy levels in the superposition of energy eigenstates. 6.7 Quantum numbers Two Hermitian matrices can share a set of eigenvectors if and only if they commute. Therefore the observable quantities associated with different operators can be known simultaneously if and only if the corresponding operators commute. We call a set of observables which can be known simultaneously quantum numbers. In general the observable properties we wish to refer to are those independent of time. Partly the reason is that time-dependent quantities will vary quickly and appear as their average values in experiments with finite resolution. Since time-independent expectation values require the operator to commute with the Hamiltonian, it is usual to take the energy as one of the observables, and therefore the quantum numbers of a particle are the observable properties associated with the set of operators which commute with the Hamiltonian. Mathematically the requirement for an observable to be a good quantum number is that the quantity is conserved: dhQ̂i =0 (237) dt where Q̂ is the operator associated to the observable. From Ehrenfest’s theorem we therefore have that Dh iE Q̂, Ĥ =0 (238) and so h i Q̂, Ĥ = 0. (239) Therefore, for an observable to be a good quantum number in a given quantum system, the corresponding operator must commute with the Hamiltonian. Note that the Hamiltonian always commutes with itself, and so energy is always a good quantum number. Note also that just because the quantum system admits a given observable as a good quantum number does not mean that all states of the system have this quantity well defined. As an example of a set of quantum numbers, consider the electron in the Hydrogen atom (see lecture 10). This can be assigned the following quantum numbers simultaneously: Lecture 6 MATRIX MECHANICS 58 PX2132: Introductory Quantum Mechanics energy: Ĥ|ni = En |ni (240) squared total angular momentum: L̂2 |li = ~2 l (l + 1) |li (241) z-projection of angular momentum: L̂z |mi = ~m|mi (242) ~ spin: Ŝ|s = ±i = ± |s = ±i (243) 2 and therefore we can write certain states of the electron in the Hydrogen atom, unambiguously, as |n, l, m, si where the letters label the quantum numbers. We have, for example, Ĥ L̂z |n, l, m, si = L̂z Ĥ|n, l, m, si = En ~m|n, l, m, si. (244) Lecture 6 MATRIX MECHANICS 59 PX2132: Introductory Quantum Mechanics 7 Quantum mechanics Videos: V7.1: Infinite dimensional Hilbert spaces V7.2: Fourier transforms V7.3: Differential operators V7.4: The postulates of quantum mechanics V7.5: Schrödinger’s cat demo Topics: Functions as infinite-dimensional vectors: examples of operators Equivalence of Schrödinger, Heisenberg, and Dirac notations expectation values wavefunction overlap The postulates of quantum mechanics interpretations of quantum mechanics For the exam you should be able to: work with functions as elements of a vector space, including taking inner products state the forms of the operators Ĥ , V̂ , p̂, and x̂ in the position basis confirm the Hermiticity of given differential operators in the position basis For the exam you will not be required to: recount details of dierent interpretations of quantum mechanics understand the dead cat. Lecture 7 QUANTUM MECHANICS 60 PX2132: Introductory Quantum Mechanics 7.1 Infinite-dimensional Hilbert spaces Functions f (x), including complex functions, obey all the axioms of linear vector spaces, and can be thought of as infinite-dimensional vectors. Recall Eq. 159, the resolution of the identity, which tells us that for a complete set of orthonormal vectors {|ei i} we have: X I= |ei ihei |. i The position eigenstates {|xi} form a complete orthonormal basis for an infinite-dimensional vector space. The resolution of the identity can be written Z ∞ I= |xihx|dx (245) −∞ or in three dimensions ZZZ ∞ I= |xihx|d3 x. (246) −∞ This is consistent with the Born rule for the probability P (x): 2 P (x) = |hx|ψi| (247) and the fact that the total proability to find the particle is 1: Z ∞ I= |xihx|dx (248) −∞ ↓ Z ∞ hψ|I|ψi = hψ|xihx|ψidx (249) −∞ ↓ Z ∞ hψ|ψi = 1 = P (x) dx. (250) −∞ Note that it’s fine to pull |ψi through the integral, as it is not itself a function of x. We will return to this in section 7.7. 7.2 Bases A useful trick, the infinite-dimensional generalisation of Eq. 178, is to act the x̂ operator on Eq. 245, using x̂|xi = x|xi (251) Lecture 7 QUANTUM MECHANICS 61 PX2132: Introductory Quantum Mechanics to give the representation of the position operator in the position basis: Z ∞ x̂ = x|xihx|dx. (252) −∞ The momentum eigenstates {|pi} also form a complete orthonormal basis for the same space, and we also have Z ∞ I= |pihp|dp (253) −∞ and Z ∞ p̂ = p|pihp|dp. −∞ The energy eigenstates solving the TISE form complete orthonormal bases: Ĥ|ni = En |ni (254) and we can similarly define X I= |nihn| (255) n and X

PX2132 Introductory Quantum Mechanics Notes (PDF)

Document Details

Tags

Related

Summary

Full Transcript

Upgrade to continue