UNIT I QUANTUM MECHANICS BY Dr Leena Gahane

UNIT –I QUANTUM MECHANICS BY Dr. Leena Gahane DEPARTMENT OF PHYSICS ACET , NAGPUR

Planck’s Quantum Postulate • Radiating body consists of large number of atomic oscillators have energy in the form of discrete unit of energy E =hν. • where, h is a new fundamental constant called as Planck’s constant : h = 6. 63 x 10 -34 Joules sec • Atomic oscillators can absorb or emit energy in discrete unit known as quanta

Einstein's Extension In 1905 Einstein modified Planks theory of radiation saying that light must be composed of energy particle called as Photons. • A light beam is regarded as a stream of photon having velocity of light ‘c’. • Higher the frequency of the wave, higher is the energy of each photon. • The intensity of a monochromatic light beam is related to the concentration of photons. I = Nhν, where N is number of photons • When photons encounter matter, they impart all their energy to the particles of matter and vanish.

Properties of Photon • Energy: E =hν • Velocity: Photons always travel with the velocity of light, ‘c’. • Mass: • Rest mass of Photon : m 0 = 0 • Linear Momentum: • Angular Momentum: S = 1ħ • Electrical nature: Photons are electrically neutral and cannot be influenced by electric and magnetic fields.

Compton effect • In 1923, while performing an experiment on x-ray scattering, A. H. Compton observed that the spectrum of scattered beam contains the shift between modified (scattered) and unmodified (unscattered) wavelength. This shift is called as Compton shift and the effect is known as Compton effect. h/λf Before After Incident photon h/λi scattered photon Φ θ Electron recoiled electron P

Conservation of energy From this, Compton derived the change in wavelength ∆λ= Maximum for Φ = 180 o ∆λ= Minimum for Φ = 0 o Conservation of momentum along X-axi Conservation of momentum along Yaxis Δλ =Compton shift; λf =modified wavelength ; λi = unmodified wavelength; h = Planck’s constant; m 0 = mass of electron; Φ = Scattering angle. Θ = angle made by recoiled electron with incident direction.

• From the figure , The wavelength of one peak does not change as the angle is varied. This is called unmodified component (λi ). • The wavelength of other peak varies strongly with angle. This is called modified component (λf ). • The change in wavelength is known as Compton shift. Δλ = λf – λi =h/m 0 c(1 -cos Φ). • Compton shift depends on the scattering angle θ, but not on the target material and the wavelength of incident radiation. Variation of intensity of Xrays as a function of wavelength, at different angle.

Limitation • Visible light can not cause Compton Effect. As the maximum Compton shift is 0. 0484 Ao Hence Compton shift is insignificant for visible radiation. Therefore Compton Effect can be observed only when, the striking photon is highly energetic. • Scattering atom should be of low atomic no. For higher atomic number , the number of loosely bound electrons are less , hence the probability of modified component is less Hence Compton shift is negligible. Therefore the intensity of unmodified component is large. Explanation on the basis of Quantum theory • Modified Component – Elastic Collision of X-ray photon with free or loosely bound electron. • Unmodified Component - Collision of X-ray photon with tightly bound electron. In this case whole atom is involved in the collision and hence m 0 (mass of electron) in the Compton shift is replaced by M (mass of atom). As M>> m 0 Compton shift is negligible. Hence unmodified components exists.

Failure of Classical Theory • According to classical wave theory, X-rays are regarded as electromagnetic waves of frequency • If these X-rays are incident on a material then it causes electron in the material to oscillate with the same frequency. • These electron reradiate with same frequency in the form of electromagnetic waves. • Hence, the scattered wave or radiation should have same wavelength as that of incident radiation. • Secondly electron radiate uniformly in all directions. • Wavelength of scattered radiation should not depend on scattering angle Φ. • Thus, Classical theory does not give the explanation of modified wavelength.

WAVE-PARTICLE DUALITY OF LIGHT • Light exhibit phenomenon of diffraction, interference etc. These can be explained on the basis of wave theory of light according to classical theory of light • The phenomenon such as photoelectric effect, Compton effect etc. can be explained on the basis of Quantum theory. According to quantum theory, light consist of particles called as photon having energy hν. • In some phenomenon light must be considered as a wave and in some phenomenon it behaves as a particle. This can be explained by following examples: • Energy of photon = E = hν • Momentum of photon = p = h/λ Here, E & p are the characteristics of particle and ν & λ are the characteristics of wave. • These equation reflect the wave-particle dualism.

De Broglie De-broglie hypothesis In 1923 Prince Louis de-Broglie postulated that as nature exhibits a great amount of symmetry , wave-particle dualism need not be the special feature of light alone but the material particle must also exhibit such dual behaviour i. e. particle as well as wave nature. The waves associated with moving particles are called matter waves or de Broglie waves, and the wavelength of matter waves are called de-Broglie wavelength. de Broglie relation Planck’s constant de-Broglie wavelength Momentum of particle • de-Broglie wavelength associated with accelerated charged particle: For electron

Properties of Matter Waves • Wavelength of matter waves is given by , λ= h/ mv. • It is seen that as mass increases, its wavelength tends to Zero. Thus the wavelength of macroscopic bodies is insignificant in comparison to the size of body even at very low velocities. • On the other hand, de-Broglie wavelength associated with microscopic particle is significant. • Matter waves are produced by the motion of particles and are independent of the charge. therefore they are neither electromagnetic nor acoustic waves but are new kind of waves. • They can travel through vacuum and do not require any material medium for there propagation. • Smaller the velocity of the particle. The longer is the wavelength of matter wave associated with it. • The velocity of matter wave depends on the velocity of material particle and is not constant quantity. • The velocity of matter wave is greater than the velocity of light. • They exhibit diffraction phenomenon as any other waves.

Bohr’s quantization condition • As electron revolves around nucleus, matter propagate along the circumference again and again. • If a wave must meets itself after going round one circumference then the wave would be in phase with itself and produced standing wave pattern. • The stationary wave pattern can form along circumference if only integral no. of wavelength fit into the orbit. Thus 2πrn =nλ = nh/mv mv rn = n ħ Ln = n ħ

Bohr’s quantization condition n= 3 n=6 2πr ≠ nλ no standing wave pattern 2πr = nλ Standing wave pattern

Davisson-Germer experiment • This experiment provided a convincing proof of the wave nature of matter and first time was studied by C. J. Davisson and L. H. Germer in 1927 • They observed the diffraction of an electron beam incident on a nickel crystal. • The Intensity of scattered electron beam measured as a function of the scattering angle Φ.

It is observed that for the accelerating voltage V = 54 V, the electron are scattered more pronouncedly at an angle of 500 with the direction of incident beam ( φ = 500) Hence, corresponding de. Broglie wavelength for electron is λ=12. 26/√ 54 =1. 66 Ǻ(experimental) Applying Bragg’s Law of X-ray diffraction for Planes that are separated by distance d, nth peak at angle θ : 2 d sin θ = nλ For Nickel, d=0. 91 Ǻ, n=1 and Φ=50 o Hence θ=65 o Wavelength for Electron is λ=1. 65 Ǻ (theoretical) Both λ (experimental) & λ (theoretical) are in close agreement. Hence, matter wave exist.

Concept of Wave packet • • • According to de-Broglie, matter waves are those waves which are associated with moving particles. Particle is entity confined to a very small volume like Wave spread over a large volume of space like Let us assume that a particle like an electron can be described by the equation This equation represent a pure sine wave that has no beginning and no end. The characteristics of wave is wavelength λ and the characteristics of particle is momentum These waves are infinitely extended and completely non localized. Hence, mono frequency wave can not represent a particle. It implies that the de-Broglie waves are not harmonic wave but the combination of several waves. So, de-Broglie waves are the superposition of several waves having slightly different frequencies. Such a wave packet possesses both wave and particle properties. It has wave like properties since it is constructed from waves It behaves as a particle as it has a particle like localization. So, Wave packet is a geometrical representation of how an object simultaneously possesses both particle and wave properties.

If Δx is smaller i. e. the wave packet is very narrow Δλ is smaller Δk is larger So, = Δx Δk ≈ 1 As, The above relation represent the lowest limit of accuracy. Therefore, We can write more generally Which is H. U. P

Synthesis of wave packet Beats are formed when two or more waves of slightly different frequencies combine.

Heisenberg Uncertainty Principle • Heisenberg uncertainty principle state that “It is not possible to measure position and momentum of particle simultaneously with high accuracy”. • Uncertainty Principle implies the limit of accuracy with which we can make measurements. • If Δx is the uncertainty in position and Δp is the uncertainty in momentum Δx Δpx ≥ ħ • We cannot have simultaneous knowledge of ‘conjugate’ variables such as position and linear momentum, energy and time, angular momentum and angular position with greatest accuracy. • So, ΔE Δt ≥ ħ Δθ ΔL ≥ ħ

HEISENBERG UNCERTAINTY EXPERIMENT (THOUGHT EXPERIMENT: Diffraction experiment ) To prove : - Δy Δpy ≥ h Uncertainty in momentum along Y-direction Heisenberg • To determine the coordinate of an electron along X-axis, place a slit of width ‘d’ perpendicular to the direction of motion of electron such that Δy ≈ d • If the slit is narrow enough then diffraction of electrons takes place so, Sinθ = λ/d • The uncertainty in momentum of electron parallel to Y-axis is given by Δpy=px. Sinθ Δpy=(h/λ)(λ/d) = h/d ≈ h/ Δy Thus Δy Δpy ≥ h

Application of Heisenberg Uncertainty Principle q Electrons can not present inside the nucleus. • Solution: - Suppose electron is present in the nucleus having radius 1 x 10 -14 m implies that maximum uncertainty in the position of electron is equal to the diameter of the nucleus. Thus, Δx = 2 x 10 -14 m. • Minimum uncertainty in its momentum is given by Δp ≥ ħ/ Δx = 5. 2 x 10 -21 Kg m/s Δp ≈ p = 5. 2 x 10 -21 Kg m/s • The minimum Energy of the electron in the nucleus is then given by Emin = pminc =(5. 2 x 10 -21 Kg m/s)(3 x 108 m/s) Emin = 1. 56 x 10 -12 J = 9. 7 Me. V • But maximum K. E. of electron or β particle = 4 Me. V • Hence, Electrons can not present inside the nucleus.

Phase velocity and Group velocity and their relations • Phase velocity vp: It is the velocity with which a definite phase of the wave propagates in the medium. vp=ω/k • Group Velocity vg: The average velocity with which group of waves having slightly different wavelength travel along the same direction is called group velocity. vg= dω/dk • vg = vp , if all constituent waves travels with same phase velocity. • vg < vp , if t he waves of different wavelengths travel in a medium with different velocities • Relation: i) vg= vp ( In non-dispersive medium when vp is not function of λ. ) ii) vg< vp ( In Dispersive medium when vp is function of λ. )

Difference between phase velocity and group velocity Phase Velocity Group Velocity • It is the velocity with which a definite • The average velocity with which phase of the wave propagates in the group of waves having slightly medium. different wavelength travel along the same direction. • vp=ω/k • vg= dω/dk • Characteristics of individual wave • Characteristics of group of waves • For non relativistic particle, vp = v/2 and for relativistic particle, vp = c 2 /v • For non relativistic particle and relativistic particle, vg = v

Concept of wave function ψ • Wave represent the propagation of disturbance in a medium. In quantum mechanics, it is difficult to derive the equation of motion. So, Schrödinger gave a mathematical function to describe equation of motion of particle called as wave function denoted by ψ. • Ψ is a function of both space (x, y, z) and time t • Ψ is a complex quantity containing real and imaginary term. i. e. Ψ = x+iy • It is not an observable quantity. Hence it does not have any physical significance. Physical interpretation of probability density • The physical quantity lψl 2 that is used to define the probability of finding the particle described by the wave function ψ at a particular time t, at a particular point (x y z) contained in the volume. • i. e. P=lψl 2=∫∫∫ψ*ψ dx. dy. dz where dx. dy. dz=dv , dv is the total volume.

Normalization Condition • The probability of finding the particle somewhere in the universe at a time is always unity. • P = ∫∫∫ψ x ψ dv = 1 Condition for well behaved wave function • Wave function should be normalized • Wave function should be single valued • Wave function should be finite and continuous and its first derivatives with respect to space coordinates must be continuous

Schrödinger’s Time dependent wave equation The Schrodinger Equation is one of the fundamental equation in Quantum Mechanics. The Schrödinger wave equation in its timedependent form for a particle of energy E moving in a potential V in one dimension is: where V = V(x, t) i = √-1 Time-Independent Wave Equation ,

Infinite Square-Well Potential • The simplest system is that of a particle trapped in a box with infinitely rigid walls that the particle cannot penetrate. This potential is called an infinite square well and is given by: • Clearly the wave function must be zero where the potential is infinite. • Where the potential is zero (inside the box), the wave function is determined from Schrödinger time-independent wave equation • The time-independent Schrödinger wave equation becomes: where • The general solution is: ∞ ∞

Quantization • Boundary conditions of the potential indictate that the wave function must be zero at x = 0 and x = L. This yields valid solutions for integer values of n such that k. L = np. • The wave function is: • The normalized wave function becomes: ∞ Thus • The same functions is applicable for a vibrating string with fixed ends. ∞

Quantized Energy • • The quantized wave number now becomes: Solving for the eigen energy yields: Eigen energy OR • The special case of n = 1 is called the ground state. • If n = 0 kn = 0 the potential well. • Note that the energy depends on integer values of n. Hence the eigen energy is quantized. Ψn = 0 The particle does not exist inside

• Quantum mechanically, The zero point energy is the consequence of the uncertainty principle. If E = 0 p=0 If p = 0 λ = infinity • If λ = infinity The particle cannot be confined to a box. • So, the particle must have certain minimum amount of kinetic energy. • Classically, the particles may have zero point energy but Quantum mechanically it is not possible.

Dependence of quantization on width of well • • • Let En and En+1 are two adjacent energy levels ΔE = En+1 – En = nh 2/4 m. L 2 Now assume two electron values for L L= 1 cm = 10 -2 m ΔE = n x 0. 74 x 10 -14 e. V This energy difference between the successive levels is so significant and electron possess thermal K. E. =10 -3 e. V. Then electron can easily move from one level to another level without any external energy. Thus the energy levels are quasicontinuous. • If L = 10Ǻ = 10 x 10 -10 ΔE = 0. 74 x n e. V • This energy difference between the adjacent level is very large and electron can not move into higher energy level without any external energy i. e. energy levels are wider • This shows that quantization of energy depends on width of the well and n

Tunneling • • • Consider a particle of energy E approaching a potential barrier of height V 0 and the potential everywhere else is zero. If the energy of the particle is greater than the potential barrier, then particle cross the barrier. Secondly, If energy of the particle is less than the potential barrier, then particle will not cross the barrier classically. But quantum mechanically, If energy of the particle is less than the potential barrier, then particle get reflected through the barrier. The penetration through barrier by quantum particle is called tunneling. Now the probability of the particle penetrate to the other side of the barrier is called transmission coefficient. It is given by: T = Probability density of transmitted wave/ Probability density of incident wave

Quantum Tunneling Incident particle ψI Transmitted particle ψIII E 0 ψIII Reflected particle T (I) (III)

• Quantum mechanics shows that the transmission coefficient is given by: T = T 0 e -2 K L = T 0 e -2 L √ 8 π 2 m ( V-E) / h Where, T 0 is a constant ≈ 1 • Above equation shows the probability of particle penetration through a potential barrier which depends on the height and width of potential barrier. Application of Tunneling: • Tunnel diode, • Scanning tunneling electron microscope • α decay