1. Carefully define or explain:

(1) Correspondence principle

(2) Uncertainty principle

(3) Probability current density

(4) Pauli’s exclusion principle

(5) Zero point energy

(6) Fermi golden rule

(7) Stark effect

(8) The postulate of special relativity

(9) Comparison of the Bose-Einstein, Maxwell-Boltzmann and Fermi-Dirac distributions

(10) Bohr’s model of the atom

(11) Franck-Hertz experiment

(12) The origin of the fine structure and hyperfine splitting of hydrogen

  1. Describe the differences between the Schrödinger’s and Heisenberg’s views of Quantum Mechanics.
  2. (a) Find the eigenfunctions and the energy spectrum of a particle in the potential well given by

(b) Find the coordinate and the momentum matrices in the energy representation of a particle in the potential well.

  1. Show that if the two Hermitian operators A and B satisfy the commutation relation AB-BA=iC, the following relation will hold:

.

  1. Prove if two operatorsA and B commute, they share common eigenstates.
  2. Find the wave functions and energy levels of the stationary states of atwo -particle plane rotator with a moment of inertia equal to I=a2, where is the reduced mass of thethis poir of particle and a is their distance apart.
  3. Two particles of mass m are attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the center (but the center point is fixed). (a) Find the allowed energies of the rigid rotor. (b) What is the degeneracy of the nth energy level?
  4. A system described by the Hamiltonian

is called an “anisotropic harmonic oscillator”. Determine the possible energies of this system, and for the isotropic case ( calculate the degeneracy of the level En.

  1. Find the energy spectrum of a system whose Hamiltonian is

,

Where a and b are small constants (the “anharmonic oscillator”).

  1. (a) Find the bound states energy spectra for an attractive potential .

(b) Scattering from a delta-function well.

A particle is moving along the x-axis. Find the probability of transmission of the particle through a delta-function potential barrier at the origin.

  1. (a) Determine the energy levels and the normalized wave functions of a particle in a “potential well”. The potential energy V of the particle is

(b) Calculate the expectation values of x, p, and the uncertainty and for ground state.

(c) Show the condition,with which the above result agrees with the corresponding classical result.

(d) Explain the fact that the ground state energy of a particle in the potential well is different from zero.

  1. Suppose we have two particles, both of mass m, in the previous infinite square well. Find the ground state and excited state wave functions and the associated energies for

(a) if the two particles are non-interacting,

(b) if the two particles are identical bosons, and

(c) if the particles are identical fermions.

  1. A particle is in the ground state in a box with sides at x=0 ad x=a. Suddenly the walls of the box are moved to , so that the particle is free. What is the probability that the particle has momentum in the range (p, p+dp)?
  2. Let s1 and s2 be the spin operators of two spin-1/2 particles. Find the simultaneous eigenfunctions of the operators s2 and sz, where s=s1+s2.show that these are also eigenfunctions of the operators .
  3. Suppose an electron is in a state, in which the component of its spin along the z-axis is +1/2. What is the probability that the component of the spin along an axis z’ (which makes an anglewith the z-axis) will have the value +1/2 and –1/2? What is the average value of the component of the spin along thez’ axis?
  4. Consider two electrons in a spin singlet state. If a measurement of the spin of one of the electrons shows that it is in a state with sy=1/2, what is the probability that a measurement of the x-component of the spin yields sx=-1/2 for the second electron?
  5. Find the possible energies of a particle in the spherical potential well given by V(r)=-V0 if r<a, and V(r)=0 if r>a.
  6. Assume that, at time t=0, the wave function x) of a particle is of the form

.

Find the change in time of this wave-packet if, for t>0, no force acts on the particle.

  1. Consider a particle subject to a constant force F in one dimension. Solve for the propagator in coordinate space.
  2. A plane rigid rotator having a moment of inertia I and an electric dipole moment d is placed in a uniform electric field E. By considering the electric field as a perturbation, determine the first non-vanishing correction to the energy levels of the rotator.
  3. A charged-particle linear harmonic oscillator is in a time-dependent uniform electric field given by , where A and  are constants. If at t=-, the oscillator is in its ground state, find, to the first order approximation, the probability that it will be in its first excited state at t=.
  4. Consider the three-dimensional infinite cubical well

(a)Find the stationary states.

(b)Find the ground state and the first excited states and their associated degeneracy.

Now let’s introduce the perturbation

(c)Find the first-order correction to the ground state energy, and

(d)Find the first excited state.

  1. Use the Gaussian trial function to obtain the upper bound-of the ground-stats energy of the one-dimensional harmonic oscillator.
  2. Find the total cross-section the scattering of slow particles by the spherical potential well
  3. A charged-particle linear harmonic oscillator is in its ground state when a time-dependent electric field is suddenly switched on. Calculate the probability of the excitation of the nth level of the oscillator, assuming that the perturbation theory is in appropriate in this case.

26.Prove if two operatorsA and B commute, they share common eigenstates.

27.Describe the differences between the Schrodinger’s and Heisenberg’s views of Quantum Mechanics.

28.Write down the Hamiltonian in Quantum Mechanics for a charged particle moving in electromagnetic field.

29.Find the bound states (E < 0) for a delta potential, V=.

30.A charged particle with mass m is constrained to move on a spherical shell in a weak uniform electric field E. Obtain the energy spectrum to second order in the field strength.

31.(a). The eigenvalue equation of the harmonic oscillator isgiven as:

We usually manage the harmonic problem in the X space with the Hermite polynomials as its wave functions,formulate this problem in number space insteadusing the following lowering and raising operators:

and

Find out and with proper coefficients.

(b). Calculate the expectation values, and .

32.(A) Solve the time-independent Schrodinger equation with the time-independent perturbation method. Find the first- and second-order corrections to the energy and the first-order correction to the wave function. If the unperturbed states are two-fold degenerate, find out the first-order correction to the energy.

(B) Consider a quantum system with just three linearly independent states. The Hamiltonian, in matrix form, is

whereis a constant and εis some small number.

(a)Write down the eigenvectors and eigenvalues of the unperturbed Hamiltonian(ε=0).

(b)Solve for the exact eigenvalues of H. Expand each of them as a power series , up to the second order.

(c)Use first- and second-order nondegenerate perturbation theory to find the approximate eigenvalue for the nondegeneratestate of unper turbed H. Comparewith the exact result from (b).

(d)Use degenerate perturbation theory to find the first-order correction to the two degenerate states of the unper turbed H. (Compare the exact results).

33.(A) For the system with a spherically symmertric potential like the Hydrogen atom, states are withspecified quantum numbers n,, and m. What are the selection rules forthe spontaneous emission?

(B) Deriveout these selection rules? Using the commutators of Lz with x, y, and z, and the following identity

34.Find the energy spectrum of a system whose Hamiltonian is

,

Where a and b are small constants (the *anharmonic oscillator*)

35.A particle is moving along the x-axis. Find the probability of transmission of the particle through a delta-function potential barrier at the origin.

36.A time t=0 a spin 1/2 particle with spin in the x-direction enters a region of space in which there is a uniform magnetic field H in the z-direction. Find the probability thatat time t the spin is still in the x-direction.

37.Consider two electrons in a spin singlet state. If a measurement of the spin of one of the electrons shows that it is in a state with sy=1/2, what is the probability that a measurement of the x-component of the spin yields sx=-1/2 for the second electron?

38.Show that if the two Hermitian operators A and B satisfy the commutation relation AB-BA=iC, the following relation will hold:

39.(a) If φi (x) and φj(x) are two different non-degenerate eigenfunctions of the time independent SchrÖdinger equation for a potential V(x). Show that

(b) Write down the form of normalized anti-symmetric total eigenfunctions of the ground states of the lithium atom

40.(a) Show that the concentration of electrons in the conduction band of an intrinsic semiconductor at temperature T is

(b) Show that the concentration of holes in the valence band of an intrinsic semiconductor at temperature T is

where kB=Boltzman constant, me=effectivemass of electron, mp=effective mass of hole, μ=chemical potential, Eg=band gap between conduction band and valence band. Note that

.

41.Consider a central potential

V(r) =

In this case, we say that we are considering a hard sphere of radius R. If the incident energy ofaparticle is low, using the method of partial wave to calculate the total scattering cross section.

42.Show that for any normalized |Ψ>,<Ψ|H|Ψ≥ E0,where E0is the ground state energy (i.e. the lowest eigenvalue). And show that if |δΨ> is a small deviation from the ground-state |Ψ>, the lowest order of the deviation of <Ψ|H|Ψ>from E0is (δΨ)2.

43.If |n> with n=0,1,2,3, .., are the eigenstates of the number operator of a one-dimensional simple harmonic oscillator, calculate the matrices of the position operator and the momentum operator based on the basis set of{ |n >}.

44.Find the uncertainty relation between, the rotation angle about the z-axis, and Lz, the z component of the angular momentum.

45.A particle is in a potential V(x)=V0sin(), which is invariant under the transformation x→x+ma, where m is an integer. Is momentum conserved? Discuss the eigenvalues and eigenstates of the one-dimensional Hamiltonian.

46.Let R(,n) be the operator that rotates a vector by about the axis n. Show that the four successive infinitesimal rotations, R(,i),R(,j),R(-,i), and lastly R(-,j) is equivalent to R(-,k). Then use this identity to show that [Lx,Ly]=iħLz. (This is called consistent test).

47.Consider a particle in a state described byΨ=N(x+y+2z)exp(-αr), where N is a normalization factor. Show that the probabilities of finding the Lz eigenstates are P(1z=0)=2/3, P(1z=+h)=1/6, and P(1z=-h)=1/6.

48.Constructthe four antisymmetrized wavefunctions of a two-electron system, whereσ1 andσ2 are the spin states of the two electrons and * is the total spin state.Ψ(r1,r2) is the orbital part of the wavefunction. Assume that the two electrons occupy the orbitals.

49.If a proton has a uniform charge distrubution of radius R, the attractive potential between the electron and the proton will be for r≤R and V(r)= for r>R. Calculate the first order shift in the ground-state energy of hydrogen. You may assume R <a0(the Bohr radius).

50.For the attractive delta function potential use a Gaussian trial function to calculate the upper bound on E0, the gound state energy.

51.When a particle is scattered from a square well potential of depth V0 and range, show that the s-wave phase shirt is , where k and k’ are the wave numbers inside and outside the well, respectively.

  1. Find the operator of a parallel displacement over a finite vector in terms of the momentum operator

From Taylor expansion,

So

  1. A particle of energy E is incident upon a potential barrier of height and width d (Fig.1). Calculate the transmissionprobability of the particle through the barrier as a function of , where. And locate maxima and minimum in the transmission probability.
  1. Start from the infinitesimal time evolution operator

Given by the path integral, prove satisfying the Schrödinger equation

  1. Calculate

Where and

  1. A spinor is rotated by 30 with respect to the rotating axis. What is the state after the rotation?
  1. In the representation ofthe eigenvectors of which is the z component of the spin operation with S=, is written as . In the same representation, another spin operator is defined by

= /

Initially, we have a statewhich is eight states of Sz with an eight value of

Now we make a measurement of .

(a)What’s the probability of getting states oftermeasurement?

(b)If we make a measurement ofright after（a）, what’s the probability of finding the states?

  1. A fru particle is restricted to m ove in if the wavefunction of this particle is

1、calculate <X>

2、calculate ,and (c) the probability of finding the particle in the region

  1. A beam of particle is scattered by a potential . Show that

where is the step function and , andarethe incident and scattered wavevectors of the particle, respectively.

  1. A systemhas a Hamiltonianof

,

where k andare constants.

What the following symmetries the system has, translation, parity, time translation and time reversal? And why? What conserved quantities the symmetries correspond with?

  1. Calculate the following expectation values,

for a simple homonic oscillator where states and > are eigenstates of with eigenvalue andrespectively.

  1. The creation and annihilation,andfor fermions which operators.

(: Identity operator), and

1、Show that the fermion number operator

Satisfies

2、If one eigenvalue of is zero, the only other eigenvalue, is=1.