Quantum Physics.docx

**QUANTUM THEORY** **Need of Quantum Mechanics** Broadly, there are two types of mechanics called classical mechanics and quantum mechanics. Classical mechanics explained successfully motion of the objects which can either be observed directly or can be made observable by instruments like microsco...

**QUANTUM THEORY** **Need of Quantum Mechanics** Broadly, there are two types of mechanics called classical mechanics and quantum mechanics. Classical mechanics explained successfully motion of the objects which can either be observed directly or can be made observable by instruments like microscope. But, the classical mechanics can not explain the mechanics of subatomic particles like electron,.proton, neutron etc. Then there comes in picture the quantum mechanics, which explain the mechanics of these subatomic particles successfully. Following examples will show that classical mechanics was inadequate to give explanation of observed facts: **(a) Photoelectric effect**: when a photon of light falls on the metal surface and if the frequency of light is more than the certain minimum frequency (called threshold frequency), then electrons are ejected from metal surface. This effect is known as photoelectric effect. Following are the experimental observations about photoelectric effect. 1\. If the frequency of the incident radiation is smaller than the metal's threshold frequency ( a frequency that depends on the properties of the metal), no electron can be emitted regardless of the radiation's intensity 2\. No matter how low the intensity of the incident radiation, electrons will be ejected instantly the moment the frequency of the radiation exceeds the threshold frequency. 3\. At any frequency above threshold frequency, the number of electrons ejected increases with the intensity of the light but does not depend on the light's frequency. 4\. The kinetic energy of the ejected electrons depends on the frequency but not on the intensity of the beam; the kinetic energy of the ejected electron increases linearly with the incident frequency. According to classical theory of light, there should be no threshold frequency, the photoelectric current should increase with increase in frequency of light and kinetic energy of ejected electrons should increase with increase in intensity of light. There must also be a finite delay between the incidence of photons and ejection of electrons from metal surface. But experimental results were totally opposite, like, photoelectric effect is instantaneous, and other observations were contradicted to aforementioned classical theory based observations. Therefore classical mechanics failed to explain the photoelectric effect. These observations successfully later on by Einstein's theory of photoelectric emission, which is based on Planck's Quantum theory of Radiation. **(b) Stability of atom:** Atomic model prepared by Rutherford was based on classical physics. This model assumes the atom to be consisting of a positively charged nucleus in the centre and negatively charged electrons revolved the nucleus in the circular orbit. Classical theory says that whenever a charged particle undergoes accelerated motion, it emits electromagnetic radiation. Therefore, an electron must emit energy continuously as its motion is accelerated. Thus, the orbital radius should decrease continuously and ultimately electron would fall in to nucleus. Therefore, an atom would collapse at the end. But, this does not happen.   As atom is stable and stability of atom could not be established by classical physics. This stability of atom was explained by Bohr's quantum theory of atom. **(c) Compton Effect:** Compton provided the most conclusive confirmation of the particle aspect of radiation. He studied the scattering X-rays by free electrons and found that the wavelength of the scattered radiation is larger than the wavelength of the incident radiation. This can be explained only by assuming that the X-ray photons behave like particles. According to classical physics, the incident and scattered radiation should have the same wavelength. (d) Spectrum of black body radiation: ------------------------------------- A black body is an ideal surface or body that completely absorbs and re-emits all the radiation that falls on it without reflection. Absorption of radiation is independent of the direction of incidence, frequency and wavelength of radiation. - It absorbs all the incident radiation irrespective of wavelength and frequency range. - There would be no radiation reflected or transmitted. - An object which is a good absorber will always be a good emitter. Therefore, a perfect black body will also be a very good emitter. - In reality, there is no perfect black body in nature. - Black Body Radiation: Electromagnetic radiation emitted by any heated object is called black body radiation. It is also called cavity radiation. - Depending on their temperature, all objects generate electromagnetic radiation. - Colder objects emit low-frequency waves such as radio or microwaves. - Heated objects emit high-frequency waves such as visible light and ultraviolet rays. ### Black Body Spectrum: One of the characteristic features of black body radiation is a continuous frequency spectrum called the Black body spectrum. It is completely dependent on temperature. Experimental Facts about Black Body Radiation --------------------------------------------- - The black body spectrum depends only on the temperature.  - The black body spectrum does not depend on the material of the body or object. - The intensity of emitted radiation increases with increase in wavelength. The Intensity becomes maximum for a particular wavelength () and then intensity decreases withy further increase in wavelength. - The wavelength corresponding to maximum intensity () of the blackbody spectrum becomes shorter as the object's temperature increases.  - Area under intensity versus wavelength curve represents total energy emitted per second per unit area by black body including all wavelengths. undefined Above is the graphical representation of Black body radiation at different temperatures. **Classical Theory fails to explain the observed spectrum of black body radiations.** **(e) Emission and absorption spectra: Classical Theory fails to explain the emissions and absorption spectra of atoms/molecules.** Origin of quantum Mechanics: Quantum theory of radiation was first proposed by Max Planck in 1901 in connection with thermal radiations emitted by black body. While studying black body radiations, he concluded that absorption and emission of thermal energy is not continuous process as in classical theory. According to him energy is emitted and absorbed by matter in discrete amounts. He assumed that matter is composed of large number of oscillating particles which can vibrate with characteristic frequency. These atomic oscillations emit or absorb energy in discrete amount defined by ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ *n* = 1,2,3 Planck's quantum hypothesis Where n is quantum number, h is Planck's constant and is frequency of oscillator. Thus we can that energy of atomic oscillator is quantized Einstein also quantized the radiant energy of light by postulating that 1. Light energy is not emitted continuously, but by discrete amount of energy which is called as quanta. 2. These quanta travel in space in the form of bundles or packets called photon which carries a definite amount of energy. Energy carried by each photon is given by where is frequency of radiation. De Broglie concept of matter waves: dual nature of matter ========================================================= In 1924, Lewis de-Broglie proposed that matter has dual characteristic just like radiation. His concept about the dual nature of matter was based on the following observations:- (a)    The whole universe is composed of matter and electromagnetic radiations. Since both are forms of energy so can be transformed into each other. (b)   The matter loves symmetry.  As the radiation has dual nature, matter should also possess dual character. According to the de Broglie concept of matter waves, the matter has dual nature. It means when the matter is moving it shows the wave properties (like interference, diffraction etc.) are associated with it and when it is in the state of rest then it shows particle properties. Thus the matter has dual nature. The waves associated with moving particles are matter waves or de-Broglie waves. **WAVELENGTH OF DE-BROGLIE WAVES** Consider a photon whose energy is given by (1) If a photon possesses mass (rest mass is zero), then according to the theory of relatively, its energy is given by (2) From (1) and (2), we have             Mass of photon Therefore Momentum of photon (3) If instead of a photon, we consider a material particle of mass 'm' moving with velocity 'v', then the momentum of the particle, p = mv. Therefore, the wavelength of the wave associated with this moving particle is given by: λ = h/p = h/mv             (4) **Wavelength of De Broglie wave is** λ = h/p = h/mv             If E is the kinetic energy of the material particle of mass m moving with velocity v, then Therefore De-Broglie wavelength 1. **De-Broglie wavelength for particle in gaseous state:** According to kinetic theory of gases, the average kinetic energy of the material particle is given by Where k=1.38 x 10^-23^ J/K is the Boltzmann's constant and T is the absolute temperature of the particle. Therefore De-Broglie wavelength **This is the De-Broglie wavelength for particle in gaseous state** **3. De-Broglie wavelength for an accelerated electron:** Suppose an electron accelerates through a potential difference of V volt. The work done by electric field on the electron appears as the gain in its kinetic energy That is E = eV h = 6.63 x 10^-34^ m = 9.1 x 10-^31^ e = 1.6 x 10^-19^ This is the de-Broglie wavelength for electron moving in a potential difference of V volt. **Velocity of de Broglie wave** To find De Broglie wave velocity, we can apply the usual formula The wavelength is simply the de Broglie wavelength, To find the frequency , we equate the quantum expression with the relativistic expression. Then we obtain The de Broglie wave velocity is therefore The wave velocity '' is the phase velocity and 'v' is the particle velocity. Because the particle vlocity *v* must be less than the velocity of light *c*, the de Broglie wave velocity is always larger than *c*. Ths means that particle lags far behind the wave which has no physical significance. Also it contradicts the special theory of relativity. Hence wave associated with a moving body cannot be represented by a single wave. **Properties of Matter waves:** 1. Matter waves are associated with moving material particle. 2. For a particle at rest v = 0, wavelength i.e a particle at rest do not exhibit wave nature. 3. Velocity of matter waves is not constant; it depends on velocity of material particle. 4. Matter waves are associated with charged as well as uncharged particle. So they are different from electromagnetic waves. 5. Matter waves are called probability waves. 6. Mater wave associated with a moving body cannot be represented by a single wave. It consist of group of waves. 7. The velocity of group of waves is called group velocity. **Davision & Germer Experiment:** The Davisson and Germer experiment confirmed the wave nature of the electrons. They demonstrated that electron beams are diffracted when they are scattered by regular atomic arrays of crystal. The experimental setup for the Davisson and Germer experiment is enclosed within a vacuum chamber. Thus the deflection and scattering of electrons by the medium are prevented. The main parts of the experimental setup are as follows:  - - - - - Classical physics predicts that the scattered electrons will emerge in all directions with only a moderate dependence of their intensity on scattering angle. ### Observations of the Davisson and Germer Experiment The intensity of the scattered electrons is not continuous. It shows a maximum and a minimum value corresponding to the maxima and the minima of a diffraction pattern produced by X-rays. The position of maxima & minima depends on the electron energy. The graphs are shown below. The method of plotting is such that the intensity at any angle is proportional to the distance of curve at that angle from the point of scattering. If the intensity were the same in all scattering angles, the curves would be circles centered on the point of scattering. It is studied from various angles of scattering and potential difference. Davisson and Germer Experiment ### ###  ### De Broglie's hypothesis suggested that electron waves were being diffracted by planes of atoms in a crystal. The basic thought behind the Davisson and Germer experiment was that the waves reflected from two different atomic layers of a Ni crystal will have a fixed [phase difference](https://www.toppr.com/bytes/wave-optics-class-12/). After reflection, these waves will interfere either constructively or destructively. Hence they produce a diffraction pattern. In a particular case, a beam of 54eV electrons was directed perpendicularly at the nickel target and the maximum scattering happens at an angle of 50^0^ with the original beam. ### The spacing between the planes is d = 0.091nm According to Bragg equation for maxima in the diffraction pattern is **d = 0.091nm, , n =1** Now we use De Broglie formula to find expected wavelength of electron For V = 54 V, we have λ = 0.166 nm  This agrees well with the observed wavelength of 0.165nm from diffraction pattern. The Davisson-Germer experiment thus directly verifies de Broglie hypothesis of the wave nature of moving bodies. **Heisenberg Uncertainty Principle** We know that the wave representation of a moving body corresponds to a wave packet or wave group. The particle that corresponds to this wave group can be found anywhere within the group at a given time. When the wave group is narrower, the particle\'s position can be specified more precisely (see fig.a) However, the wavelength of the waves in a narrow packet is not well defined. The reason is that the range of wavelengths of individual waves is large. Since wavelength λ = h/p, so the particle\'s momentum is not a precise quantity. If we make a series of momentum measurements, we will find a broad range of values. When the wave group is wider, the particle\'s wavelength can be specified more precisely (see fig. b). Therefore, the momentum can be measured more precisely. However, since the wave group is wide, the position of the particle is not well defined. If we make a series of position measurements, we will find a broad range of values. Thus we have the uncertainty principle **Statement**: According to Heisenberg uncertainty principle, it is impossible to measure the exact position and momentum of a particle simultaneously within the wave packet. If be the uncertainty in the measurement of position and be the uncertainty in the measurement of position, then Proof:-Consider a wave packet formed by the superposition of two waves that have the same amplitude '*A'* but differ by a small amount in angular frequency and a small amount in wave number: The resultant wave group is then given by **(1)** Now as and are very small quantities, so we can write Equation (1) becomes **(2)** The above equation represents a wave of angular frequencyand wave number *k* whose amplitude is modulated by an angular frequency and a wave number The amplitude of resultant wave group is given by At nodes, the amplitude of wave packet is zero. Hence Now 2A cannot be equal to zero, so If and are the positions of two consecutive nodes. then (3) (4) Subtract equation (3) from (4) is the uncertainty in the measurement of position of a particle (5) We know that , where is the uncertainty in the measurement of momentum along x-axis Equation (5) becomes If wave packet consist of number of waves, then This is the expression of the **Heisenberg uncertainty principle** **Wave function and its physical significance** If there is a wave associated with a particle, then there must be a function to represent it. This function is called wave function. Wave function is defined as that quantity whose variations make up matter waves. It is a complex quantity which consists of real and imaginary parts. Ψ=A+iB **PHYSICAL SIGNIFICANCE (BORN'S INTERPRETATION)** The wave function describes the amplitude of the wave and amplitude can be negative as well as positive. But negative value of probability is meaningless; it always lies between 0 and 1. Hence wave function ψ itself has no physical significance. The square of absolute value of wave function is always positive. According to Max Born, the square of absolute value of wave function \|ψ^2^\| represents the probability of finding the particle at a particular point and at a particular time. The wave function ψ(x,t) is a quantity such that the product P(x,t) = ψ^\*^(x,t)ψ(x,t) is the probability per unit length of finding the particle at the position x at time t. P(x,t) is the probability density and ψ^\*^(x,t) is complex conjugate of ψ(x,t) **NORMALIZATION CONDITION** The probability per unit length of finding the particle at position x at time t is P = ψ^\*^(x,t)ψ(x,t) So, probability of finding the particle in the length dx is Pdx = ψ^\*^(x,t)ψ(x,t)dx Total probability of finding the particle somewhere along x-axis is ∫Pdx = ∫ ψ^\*^(x,t)ψ(x,t)dx If the particle exists, it must be somewhere on the x-axis. So the total probability of finding the particle must be unity i.e. ∫ψ^\*^(x,t)ψ(x,t)dx=1                               (1) This is called the normalization condition. So a wave function ψ(x,t) is said to be normalized if it satisfies the condition (1). The wave function is known as normalized wave function. **ORTHOGONAL WAVE FUNCTIONS** Consider two different wave functions ψ~m~ and ψ~n~ such that both satisfies [Schrodinger equation](http://www.winnerscience.com/quantum-physics/time-independent-schrodinger-wave-equation/).These two wave functions are said to be orthogonal if they satisfy the conditions. ∫ ψ~n~^\*^ (x,t) ψ~m~(x,t) dV=0 for n≠m\]                          ( 1) ∫ ψ~n~^\*^ (x,t) ψ~m~(x,t) dV=0 for m≠n \] If both the wave functions are simultaneously normal then ∫ ψ~m~ ψ~m~^\*^ d V=1=∫ψ~n~ψ~n~^\*^ dV                                   (2) **Orthonormal wave functions:** The sets of wave functions, which are both normalized as well as orthogonal are called orthonormal wave functions. ∫ψ^\*^~m~ψ~n~dV={ 0 if   m≠n = \[1 if m=n **TIME DEPENDENT SCHRODINGER WAVE EQUATION** In quantum mechanics, the wave function ψ corresponds to the variable y of wave motion. We know that the wave function for a particle is given by Ψ(x,t) = A exp\[-i(ωt-kx)\] Put and                                     (1) where E is total energy and p is momentum of the particle. Put , we have (2) This equation is a description of the wave equivalent of a free particle moving in the +ve x- direction. But generally we are interested in situations where particle is not free i.e it is subjected to some external force. Differentiating equation (2) w.r.t. x Again differentiating wr.t. x   (3) Also on differentiating (2) wr.t. t ,we get (4) When the particle is acted upon by a force then its total energy is the sum of Kinetic and potential energies i.e. Total energy = Kinetic energy + potential energy E=p^2^/2m +V                                                         (5) Putting equations (3) and (4) in equations (5), we get (6) Which is time dependent form of Schroedinger wave equation In three --dimensional form where the particle potential V is a function of x,y,z and t. Any restriction on the particle motion will effect the potential energy V. once V is known, Schrödinger equation may be solved for the wave function Ψ of the particle form where Ψ^2^ may be determined. Time Independent Schrodinger Wave Equation ========================================== In most of the problems in quantum mechanics, the potential energy is independent of time. for such situations, wave function can be rewritten as (1) where is a function of position only Differentiating equation (1) w.r.t.x, twice , we have (2) Also on differentiating equation (1) w. r. t. t, we have (3) Put equations \[1-3\] in time dependent Schrodinger wave equation , we get +V Dividing throughout by , we have +V +V +(V- E)= 0 (4) Which is time independent form of Schrodinger wave equation in one dimension. In three-dimensional form: **Eigen wave functions and Eigen values** Such an equation, where the operator, operating on a function, produces constant times the same function, is called an eigenvalue equation. The function is called an eigenfunction, and the resulting constant numerical value is called the eigenvalue. Here operator is operating on function '*f'* produce the same function multiplied some constant value '*c*'. The function '*f'* is called eigen function and *'c'* is called eigen value corresponding to operator Consider time independent Schrödinger wave equation =================================================== In one dimension Here operator is called Hamiltonian operator. is an eigen value equation. is wave function and E is energy eigen values corresponding to Hamiltonian operator. Application of Schrodinger wave equation: Particle in a box =========================================================== Consider one dimensional closed box of width L. A particle of mass 'm' is moving in a one-dimensional region along x-axis specified by the limits x=0 and x=L as shown in fig. The potential energy of particle inside the box is zero and infinity elsewhere. Potential energy function can be represented as The one-dimensional [time independent Schrodinger wave equation](http://www.winnerscience.com/quantum-physics/time-independent-schrodinger-wave-equation/) is given by Inside the box, V(x) = 0, so the Schrodinger equation in this region becomes                                          (2) Where K is called the Propagation constant of the wave associated with particle and it has dimensions reciprocal of length. The general solution of equation (2) is Ψ=A sin Kx + B cos K x                                   (4) Where A and B are arbitrary conditions and these will be determined by the boundary conditions. \(ii) **Boundary Conditions** The particle will always remain inside the box because of infinite potential barrier at the walls. So the probability of finding the particle outside the box is zero outside the box. We know that the wave function must be continuous at the boundaries of potential well (5) (6) **(iii) Determination of Energy of Particle** Apply Boundary condition (5) to equation (4) 0=A sin (K\*0) +B cos (K\*0) 0= 0+B\*1 B=0                                                             (7) Therefore eq.(4) becomes Ψ(x) = A sin Kx                                    (8) Applying the boundary condition (6) to equation (8) ,we have 0=A sin KL (9) 'A, Cannot be zero in eq. (9) because then both A and B would be zero. This will give a zero wave function everywhere which means particle is not inside the box. Sin KL=0 (10) Where    n= 1, 2, 3 -- - - A Cannot be zero in eq. (9) because then both A and B would be zero. This will give a zero wave function every where which means particle is not inside the box. **Eigen Wavefunction of a particle in a box:** Substitute the value of K from equation (10) in equation (8) to get As the wave function depends on quantum number 'n' so we write it for 0 \