Experimental Physics Module 1: Quantum Physics Notes PDF

Course Code: SESH1130 Course Name: Experimental Physics Module 1: Quantum Physics: Introduction; De Broglie hypothesis of matter waves; Properties of matter waves; Phase velocity and group velocity and their relation; Heisenberg uncertainty principle; non-existence of electron in nucleus; Wave func...

Course Code: SESH1130 Course Name: Experimental Physics Module 1: Quantum Physics: Introduction; De Broglie hypothesis of matter waves; Properties of matter waves; Phase velocity and group velocity and their relation; Heisenberg uncertainty principle; non-existence of electron in nucleus; Wave function; Physical interpretation of wave function; Schrodinger‘s time dependent wave equation; time independent wave equation; Quantum Computing (overview). 1. IntroductIon Classical Mechanics:- It is the branch of physics, which explains the motion of macroscopic objects. It was introduced by sir Issac Newton. Quantum Mechanics:- It is the branch of physics which explains about the motion of microscopic particles like electrons, protons etc. It was introduced by Max Planck in 1990. We know that classical mechanics can successfully explain the motion of astronomical bodies (such as stars, planets satellites etc.) means Newton’s law of motion, as well as macroscopic bodies (such as motion under). Except this motion of charged particles in electromagnetic fields, elastic vibrations in solids, propagation of sound waves in glass etc. can also be explained successfully by classical mechanics, but some phenomenon like black body radiation, photo-electric effect, Compton effect, emission and absorption of light etc. could not be explained, which is explained by quantum mechanics. 1 Waves & Particles: Wave:- A wave is nothing but spreading of disturbance in a medium. The characteristics/properties of waves are 1)Amplitude 2) Time period 3) Frequency 4) Wavelength 5)Phase 6)Intensity. Particle:- A particle is a point in space which has mass & occupies space or region. The characteristics/properties of a particle are 1) Mass 2) velocity 3) Momentum 4) Energy etc. Wave Particle Duality: Light obeys the phenomena of interference, diffraction, polarization, photoelectric effect, Compton Effect etc. The phenomena of interference, diffraction and polarization can be explained by assuming that light is a form of wave. By the wave theory of light, it has been proved that light possesses a wave nature. However, some other phenomena like black body radiation, photoelectric effect, Compton Effect and emission and absorption of light can be explained only with the help of the quantum theory of light. According to the quantum theory, light radiation travels in the form of energy bundles (small packets) having energy hν called photon or quanta of energy, where ν is the frequency of radiation. Hence according to the quantum theory, light possesses a corpuscular (particle) nature. Therefore, sometimes light obeys the wave theory and sometimes the corpuscular theory. Hence, light has dual nature. 2. de-BroglIe HypotHesIs or de-BroglIe Matter Waves: Louis De-Broglie suggested that the dual nature is not only of light, but each moving material particle has the dual nature. He assumed a wave should be with each moving particle which is called the matter waves. Although these waves can travel through vaccum like electromagnetic waves, but these are different from waves, because these waves associated with all types of charged and neutral particles. 2 Definition: The waves associated with a material particle are called as matter waves or De-Broglie-waves. Wave nature of matter waves is verified by Davisson & Germer experiment, G. P. Thomson experiment etc. Particle nature of matter waves is verified by photo-electric effect, Compton effect etc. Difference between matter waves and electromagnetic waves:- Matter waves Electromagnetic waves (Light Waves) 1) Matter wave is associated with 1) Oscillating charged particle gives moving particle. rise to EM wave 2) Wavelength of matter wave is 2) Wavelength of matter wave is 𝒉 𝒉𝑪 given by 𝝀 =. given by 𝝀 =. 𝒎𝒗 𝑬 3) Wavelength of matter wave 3) Wavelength of an depends upon mass of the particle electromagnetic wave depends & velocity. upon the energy of the photon. 4) It can travel with a velocity 4) It can travel with a velocity equal greater than the velocity of light in to the speed of light in vacuum i.e. vacuum. 𝑐 = 3 × 10 𝑚/𝑠. 5) It is not an EM wave. 5) Electric field and Magnetic field oscillate perpendicular to each other & generate EM wave. 3 De-Broglie Hypothesis:- (1) The universe consists of matter and radiation(light) only (2) Matter waves also exhibit dual nature like radiation. (3) The waves associated with the material particles are called as de- broglie-waves or matter waves & the wave length associated with matter waves are called as de-broglie wave-length or matter wave-length (λ). (4) De-Broglie wave-length is given by, 𝒉 𝒉 𝝀= = 𝒑 𝒎𝒗 3. propertIes or cHaracterIstIcs of Matter Waves or de- BroglIe Waves:-  Lesser the mass of the particle, greater is the wavelength associated with it.  Smaller the velocity of the particle, longer is the wavelength associated with the particle.  Matter waves produced when the particles in motion are charged or uncharged.  Matter wave are not electro-magnetic waves.  Matter waves travel faster than the velocity of light.  Wave nature of matter gives an uncertainty in the position of the particle. 4. pHase and group velocIty Phase velocity and group velocity are two important and related concepts in wave mechanics. Waves can be in the group and such groups are called wave packets, so the velocity with a wave packet travels is called group velocity. The velocity with which the phase of a wave travels is called phase velocity. 4 Phase Velocity (vphase or vp): A point marked on a wave can be regarded as representing a particular phase for the wave at that point. The velocity with which such a point would propagate is known as phase velocity or wave velocity. Or We refer to the speed of each original wave as the phase velocity. It is represented by 𝜔 𝑣 𝑜𝑟 𝑣 = 𝑘 where, ω is angular frequency and k is the propagation constant or wave number. Group Velocity (vgroup or vg): The velocity with which the resultant envelops of the group of waves travel is called group velocity. 5 Or We refer to the speed of the resultant pattern as the group velocity. It is represented by 𝑑𝜔 𝑣 𝑜𝑟 𝑣 = 𝑑𝑘 4.1 relatIon BetWeen pHase and group velocIty: As we know the phase velocity is represented by 𝜔 𝑣 = 𝑘 (1) and group velocity is represented by 𝑑𝜔 𝑣 = 𝑑𝑘 (2) Equ. (1) can be rewritten as 𝜔 = 𝑘. 𝑣 (3) Differentiating Equ. (3) with respect to k, 𝑑𝜔 𝑑𝑘 𝑑𝑣 =. 𝑣 + 𝑘. 𝑑𝑘 𝑑𝑘 𝑑𝑘 𝑑𝜔 𝑑𝑣 ∴ = 𝑣 + 𝑘. 𝑑𝑘 𝑑𝑘 (4) Now from Equ.(2), we know that 𝑣 = , thus Equ. (4) reduces to 6  𝑑𝑣 𝑣 = 𝑣 + 𝑘. 𝑑𝑘 (5) Modifying Equ.(5) to 𝑘. 𝑑𝑣 𝑑𝜆 𝑣 =𝑣 +. 𝑑𝜆 𝑑𝑘 (6) As we know, wave number is given by 𝑘 = or 2𝜋 𝜆= 𝑘 (7) Differentiating Equ. (7) with respect to k, we get 𝑑𝜆 2𝜋 =− 𝑑𝑘 𝑘 (8) 𝑑 1 𝑑 1 ∵ 𝑥 = 𝑛𝑥 , 𝑡ℎ𝑢𝑠 ⇒ 𝑘 = −1(𝑘 ) = −1𝑘 =− 𝑑𝑥 𝑘 𝑑𝑥 𝑘 Substituting Equ. (8) in Equ. (6) 𝑘. 𝑑𝑣 −2𝜋 𝑣 =𝑣 +. 𝑑𝜆 𝑘 𝑑𝑣 2𝜋 ∴𝑣 =𝑣 −. 𝑑𝜆 𝑘 (9) Now substituting Equ. (7) in Equ. (9) 𝑑𝑣 𝑣 =𝑣 −𝜆 𝑑𝜆 This is the relation between group velocity and phase velocity. The group velocity is directly proportional to phase velocity. This means- 7  When group velocity increases, proportionately phase velocity will also increase.  When phase velocity increases, proportionately group velocity will also increase. 5. HeIsenBerg uncertaInty prIncIple:- In 1927 Heisenberg proposed “the uncertainty principle”. This Principle is a result of the dual nature of matter. “It is impossible to determine simultaneously the position and momentum of the particle with any desired accuracy.” or "The position and momentum of the particle cannot be determined simultaneously with highest accuracy. " This definition is known as Heisenberg’s uncertainty principle. If ′Δx′ & ′Δp′ are uncertainties in the measurement of position & momentum of the particle then mathematically this uncertainties of this physical variables is written as, h Δ𝑥. Δp ≥ 4π (1) Explanation: (i) If Δ𝑥 = 0, that is, the position of a particle is measured accurately, then from Equ. (1) h Δp = Δ𝑥. 4π h Δp = =∞ 0 It means that, the momentum of the particle can't be measured. 8 (i) If Δ𝑝 = 0, that is, the position of a particle is measured accurately, then from Equ. (1) h Δ𝑥 = Δp. 4π h Δ𝑥 = =∞ 0 It means that, the position of the particle can't be measured. From the above said observations made by Heisenberg, it clearly states that it is impossible to design an experiment to prove the wave & particle nature of matter at any given instant of time. If one measures position or momentum accurately, then there will be an uncertainty in the other. Thus, the Heisenberg’s uncertainty principle gives the probability of determining the particle at any given instant of time in place of certainty. 5.1 applIcatIon of HeIsenBerg's uncertaInty prIncIple:- (i) Non-existence of an electron in the nucleus (Electron cannot reside in the nucleus) Heisenberg's uncertainty principle is discussed and given in Equ. (1) above, that is h Δ𝑥. Δp ≥ 4π The radius of a typical atomic nucleus is about 5 × 10 𝑚. If the electron is present inside the nucleus, then the uncertainty in its position is atmost equal to the diameter of the nucleus, i.e., Δ𝑥 = 10 𝑚. Then, from Equ.(1), h Δp ≥ Δ𝑥. 4π 9  6.625 × 10 ∴ Δp ≥ 10 × 4 × 3.14 m ∴ Δp ≥ 0.527 × 10 kg. s is the uncertainty in the momentum of the electron. Then the momentum of the electron must atleast be equal to the uncertainty in the momentum i.e., p = 0.527 × 10 kg. The energy 'E' of the electron is given by 𝑝 𝐸= 2𝑚 substituting the value for p and m, we get (0.527 × 10 ) 𝐸= 2 × 9.1 × 10 × 1.6 × 10 0.277729 × 10 ∴𝐸= 29.12 × 10 ∴ 𝐸 = 0.009537 × 10 ∴ 𝐸 = 95.37 × 10 ∴ 𝐸 = 95.37 𝑀𝑒𝑉 as 10 𝑒𝑉 = 1 𝑀𝑒𝑉 This means that in order that an electron may exist inside the nucleus, its kinetic energy must be greater than or equal to 95.4 MeV. But experiments show that the electrons emitted by certain unstable nuclei 10 never have more energy than 3 MeV to 4 MeV. From this we conclude that electron cannot exist within the nucleus. (ii) Existence of proton and neutron inside the nucleus As above here also we start with Δ𝑥 = 10 𝑚 which is the maximum uncertainty in position. Thus, p = 0.527 × 10 kg. Here the mass of the proton and neutron is 1.67 × 10 kg. Thus substituting this in 𝑝 𝐸= 2𝑚 (0.527 × 10 ) 𝐸= 2 × 1.67 × 10 × 1.6 × 10 0.277729 × 10 ∴𝐸= 5.344 × 10 ∴ 𝐸 = 0.05197 × 10 ∴ 𝐸 = 51.97 × 10 × 10 ∴ 𝐸 = 51.97 × 10 ∴ 𝐸 = 51.97 keV as 10 𝑒𝑉 = 1 𝐾𝑒𝑉 This value of energy is smaller than energy of the particle emitted by the nucleus. So the particle like proton, neutron or heavier than that can only exist inside the nucleus. 11 6. Wave functIon and Its sIgnIfIcance:- Wave Function: The probability that a particle found at a given place in space at a given instant of time is characterized by the function Ψ(𝑥, 𝑦, 𝑧, 𝑡) and is called the wave function. Wave function can be either real or complex. Extra Note: 2 + 3i is a complex number, where 2 is a real number and 3i is an imaginary number Properties of Wave Function: 1. The wave function Ψ is in general a complex function. Ψ ∗ is its complex conjugate. 2. The wave function must be well defined, i.e., it must be single valued and continuous everywhere. 3. It is used to describe the motion of an atomic particle and is a function of both position and time. 4. Ψ represents probability amplitude, |Ψ| represents probability density i.e. probability per unit length in one dimension and probability per unit volume in three dimension. 7. pHysIcal sIgnIfIcance of Wave functIon: The wave function Ψ has no physical significance by itself. The only quantity having a physical meaning is the square of its magnitude P called the probability density. 𝑃 = |Ψ| = ΨΨ∗ where Ψ∗ is the complex conjugate of Ψ. The probability of finding a particle in a volume dx, dy, dz is |Ψ| dx. dy. dz. Further, since the particle (say electron or photon) is certainly to be found somewhere in space, the triple integral extending over all possible values of x, y, z. |Ψ| dx. dy. dz = 1 A wave function Ψ satisfying the above relation is called a normalized wave function. 12 8. scHrödInger’s Wave equatIon:- Schrödinger describes the wave nature of a particle in mathematical form and is known as Schrödinger wave equation (SWE). There are two types of SWE. (i) Schrödinger Time independent wave equation (ii) Schrödinger Time dependent wave equation (i) Schrödinger Time independent wave equation:  In 1926, Erwin Schrödinger formulated non-relativistic wave mechanics. This not only explains the mechanics of electrons and other such particles, but also their interaction with radiant energy.  He derived this equation on the lines of Hamiltonian mechanics and is called as Schrödinger Time independent wave equation. Extra Note: Relativistic approach: predicts the behaviour of particles at high energies and velocities comparable to the speed of light. Non-relativistic approach: predicts the behaviour of particles at normal energies and velocities lower to the speed of light. Hamiltonian mechanics: Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time.  When an electron or a similar particle of mass 'm' moving with a velocity 'v' in positive x-direction, it is then associated with a matter wave moving along with the particle with the same speed and is represented by Ψ = 𝐴. 𝑒 ( ) (1) Since 𝐸 = ℎ𝜐 =. 2𝜋𝜐 = ℏ𝜔 (as ℏ = 𝑎𝑛𝑑 𝜔 = 2𝜋𝜈) 13  𝐸 𝜔= ℏ (2) ℏ 𝜆= = (since ℏ = ⟹ ℎ = 2𝜋ℏ) Rearranging 2𝜋 𝑝 = 𝜆 ℏ 𝑝 2𝜋 𝑘= (𝑎𝑠 𝑘 = ) ℏ 𝜆 (3) Substituting Equ.(2) & (3) in Equ.(1), the wave function takes the form: Ψ = 𝐴. 𝑒 ℏ ℏ (4) Equ. (4) describes the wave equivalent of an unrestricted particle of total energy 'E' and momentum 'p' moving in the positive x-direction. Differentiating the above wave function in Eq.(4) with respect to 'x' twice we have 𝜕Ψ 𝑝 = 𝐴. 𝑖.. 𝑒 ℏ ℏ 𝜕𝑥 ℏ 𝜕 Ψ 𝑝 = 𝐴. 𝑖..𝑒 ℏ ℏ 𝜕𝑥 ℏ 𝜕 Ψ 𝑝 ∴ =𝑖 𝐴. 𝑒 ℏ ℏ 𝜕𝑥 ℏ 𝜕 Ψ 𝑝 ∴ =− Ψ 𝜕𝑥 ℏ 14 as 𝐴. 𝑒 ℏ ℏ = Ψ from Equ.(4). Now rearranging, we get, 𝜕 Ψ 𝑝 Ψ = −ℏ 𝜕𝑥 ℎ 𝜕 Ψ ∴𝑝 Ψ=− 2𝜋 𝜕𝑥 (5) Following Hamiltonian, Schrödinger described that the particle has total energy E, which is partly kinetic and partly potential. 1 ∴ 𝐸 = 𝑚𝑣 + 𝑉 2 (6) Modifying 𝑚𝑣 by multiplying and dividing by 'm', we get 1 𝑚 𝑣 (𝑚𝑣) 𝑝 𝑚𝑣 = = = 2 2𝑚 2𝑚 2𝑚 as p = mv Thus Equ.(6) becomes 𝑝 𝐸= +𝑉 2𝑚 𝑝 𝐸−𝑉 = 2𝑚 𝑝 = 2𝑚(𝐸 − 𝑉) (7) The wave packet of matter wave associated with this particle has the same energy and momentum of the particle (electron). Hence substituting Eq. (7) in Equ. (5) 15  ℎ 𝜕 Ψ 2𝑚(𝐸 − 𝑉)Ψ = − 2𝜋 𝜕𝑥 Or ℎ 𝜕 Ψ 2𝑚(𝐸 − 𝑉)Ψ = − 4𝜋 𝜕𝑥 𝜕 Ψ 8𝜋 𝑚(𝐸 − 𝑉) ∴ =− Ψ 𝜕𝑥 ℎ Or 𝜕 Ψ 8𝑚𝜋 (𝐸 − 𝑉) ∴ + Ψ=0 𝜕𝑥 ℎ (8) For free particle, V = 0 𝜕 Ψ 8𝑚𝜋 𝐸 + Ψ=0 𝜕𝑥 ℎ (9) This is one dimensional time independent Schrödinger wave equation. (ii) Schrödinger Time dependent wave equation:  Again when an electron or a similar particle of mass 'm' moving with a velocity 'v' in positive x-direction, it is then associated with a matter wave moving along with the particle with the same speed and is represented by Ψ = 𝐴. 𝑒 ( ) (1) Since 𝐸 = ℎ𝜐 =. 2𝜋𝜐 = ℏ𝜔 (as ℏ = 𝑎𝑛𝑑 𝜔 = 2𝜋𝜈) 16  𝐸 𝜔= ℏ (2) ℏ 𝜆= = (since ℏ = ⟹ ℎ = 2𝜋ℏ) Rearranging 2𝜋 𝑝 = 𝜆 ℏ 𝑝 2𝜋 𝑘= (𝑎𝑠 𝑘 = ) ℏ 𝜆 (3) Substituting Equ.(2) & (3) in Equ.(1), the wave function takes the form: Ψ = 𝐴. 𝑒 ℏ ℏ (4) Equ. (4) describes the wave equivalent of an unrestricted particle of total energy 'E' and momentum 'p' moving in the positive x-direction. Differentiating the above wave function in Eq.(4) with respect to 'x' twice we have 𝜕Ψ 𝑝 = 𝐴. 𝑖.. 𝑒 ℏ ℏ 𝜕𝑥 ℏ 𝜕 Ψ 𝑝 = 𝐴. 𝑖..𝑒 ℏ ℏ 𝜕𝑥 ℏ 𝜕 Ψ 𝑝 ∴ =𝑖 𝐴. 𝑒 ℏ ℏ 𝜕𝑥 ℏ 𝜕 Ψ 𝑝 ∴ =− Ψ 𝜕𝑥 ℏ 17 as 𝐴. 𝑒 ℏ ℏ = Ψ from Equ.(4). Now rearranging, we get, 𝜕 Ψ 𝑝 Ψ = −ℏ 𝜕𝑥 (5) Differentiating the above wave function in Eq.(4) now with respect to 't' once we have 𝜕Ψ 𝐸 = 𝐴. −𝑖.. 𝑒 ℏ ℏ 𝜕𝑡 ℏ 𝜕Ψ −𝑖𝐸 = Ψ 𝜕𝑡 ℏ as 𝐴. 𝑒 ℏ ℏ = Ψ from Equ.(4). Now rearranging, we get, ℏ 𝜕Ψ 𝐸Ψ = − 𝑖 𝜕𝑡 (6) At speeds small compared with that of light, the total energy E of a particle is the sum of its kinetic energy and its potential energy U, where U is in general a function of position 'x' and time 't': 𝑝 𝐸= + 𝑈(𝑥, 𝑡) 2𝑚 (7) Multiplying both sides of Eq. (7) by wave function Ψ gives 𝑝 𝐸Ψ = + 𝑈(𝑥, 𝑡). Ψ 2𝑚 (8) Now we substitute values from Eq.(5) & Eq.(6) in Eq.(8) 18  ℏ 𝜕Ψ ℏ 𝜕 Ψ − =− + 𝑈(𝑥, 𝑡). Ψ 𝑖 𝜕𝑡 2𝑚 𝜕𝑥 ℏ. 𝑖 𝜕Ψ ℏ 𝜕 Ψ − =− + 𝑈(𝑥, 𝑡). Ψ 𝑖. 𝑖 𝜕𝑡 2𝑚 𝜕𝑥 𝜕Ψ ℏ 𝜕 Ψ 𝑖ℏ =− + 𝑈(𝑥, 𝑡). Ψ 𝜕𝑡 2𝑚 𝜕𝑥 (9) This is one dimensional time dependent Schrödinger wave equation. In three dimensional time dependent Schrödinger wave equation is given by 𝜕Ψ ℏ 𝜕 Ψ 𝜕 Ψ 𝜕 Ψ 𝑖ℏ =− + + + 𝑈(𝑥, 𝑦, 𝑧, 𝑡). Ψ 𝜕𝑡 2𝑚 𝜕𝑥 𝜕𝑦 𝜕𝑧 (10) 9. quantuM coMputIng (overvIeW): Evolution of Classical computers 19 What is Quantum Computer? A quantum computer is a machine that performs calculations based on laws of quantum mechanics, which is the behaviour of particles at the sub-atomic level. Basic concept of Quantum computers & differences with existing computers:  In existing computers, all information is expressed in terms of 0s and 1s, and the entity that carries such information is called a 'bit'.  A bit can be in either a 0 or 1 state at any one moment in time.  A quantum computer, on the other hand, uses a 'quantum bit' or 'qubit' instead of a bit.  A qubit also makes use of two states (0 and 1) to hold information, but in contrast to bit. A qubit can take on the properties of 0 and 1 simultaneously at any one moment.  Accordingly, two qubits in this state can express the four values of 00,01,10 and 11 all at one time. Applications of Quantum computers  Cryptography  Artificial Intelligence  Teleportation  Quantum communication 20 Difference between Quantum computing and Classical computing -----XXXXX END OF MODULE 01 XXXXX--------- 21