Quantum Theory Ii

The IUG QUANTUM THEORY 22/6/2016 Dept. of Physics Final Exam. Time: 2 hrs.

Answer the following questions:

[20 pts] Q1) The Hamiltonian of the two dimensional isotropic harmonic oscillator is 1 2 2 1 2 2 2 H  ( p  p )   (x  y ) . o 2 x y 2 Introducing the raising operators and their complex conjugate, the lowering operators ( a and a† ) as  1  1 a  x  i p and a  y  i p x 2 2 x y 2 2 y † † a) Prove that Ho  axax  ayay 1

b) By expressing the angular momentum L in terms of a and a† , prove that L is

† conserved. You are given ai , a j  ij

H n n   n n    n  n 1 c) If o x y nxny x y , prove that nxny  x y 

[20 pts] Q2) A particle of mass m is constrained to move between two concentric impermeable spheres of radii r = a and r = b, i.e.,

V=0 0 a  r  b a V (r)    elesewhere

Find the ground state energy and normalized wave function.. By letting b U r Rr  the radial equation reduces to r

2 d 2U ll 12  x 1   V r U r  EU r sin 2 ax dx   sin 2ax 2  2        2 dr  2r  2 4a

[20 pts] Q3) Refering to Q1, a perturbation is added such that the Hamiltonian becomes H  Ho  gxy . Using perturbation method, find the first order correction to the energy eigenvalue and to the eigenfunction of the second excited state n  nx  n y  1 20 pts] Q4) The Hamiltonian of the one dimensional harmonic oscillator is

2 d 2 1 H     2 x 2 . 2 dx2 2

2 Using the trial wavefunction x  Nex with N is the normalization constant and  is a variational parameter, find the minimized ground state energy.

 2  2 2x dx 1  2 2x dx 1  You are given  e  2 &  x e  0 2 0 8 2

[20 pts] Q5) The Hamiltonian of the central force problem in a strong external magnetic field B (the Pascen-Back Effect) is given by

e   e   H  H  L  B  S  B cf 2c c

Assuming B is taken to be along the z-axis, find the ground state energy of such a Hamiltonian.

You are given: n,l,ml ,ms H cf n,l,ml ,ms   nl

Lz l,ml  ml l,ml

S z s,ms  ms s,ms The IUG QUANTUM THEORY 31/5/2015 Dept. of Physics Final Exam. Time: 2 hrs.

Answer the following questions: p2 [20 pts] Q1) Show that for any Hamiltonian of the form H  x  axn the following relation 2m

holds xpx ,H i2T  nV 

[20 pts] Q2) Suppose that an electron is in a state described by the wave function 1 r  ei sin  cos Rr 4 a) Find the eigen-value of L2. b) Find the expectation value of Lz. c) Find Lr Hint: 3 Y 0 , cos 1 4 3 Y 1,  sinei 1 8

L l,m   ll 1 mm 1 l,m 1

[20 pts] Q3) Consider a particle in a spherical potential well of radius a, i.e., 0 r  a V (r)    elesewhere U r Find the normalized wave function of the ground state energy (n=1) . By letting Rr  the radial r equation reduces to

2 d 2U ll 12  x 1   V r U r  EU r sin 2 ax dx   sin 2ax 2  2        2 dr  2r  2 4a [20 pts] Q4) Using First order perturbation theory, calculate the energy of the first two states of a aprticle confined in an infinite potwntial well od width a, whose portion AB has been sliced off such that the line OA is a straight line, as shown. The unperturbed eigenfunctios and the corresponding eigenvalues are:

2   22  0 r  sin x E 0  1 a a 1 2a 2 2 2 2 22  0 r  sin x E 0  2 a a 2 a 2

[20 pts] Q5) Consider a particle with the Hamiltonian

1 2 2 1 2 2 H  H  gxy with H  ( p  p )  k(x  y ) , o o 2 x y 2 is the Hamiltonian of the two-dimensional, isotropic harmonic oscillator. Using perturbation method, find the first order correction to the energy eigenvalue of the second excited state (n=1).   with x  a†  a  and p  i a †  a 2 2 The IUG QUANTUM THEORY 5/1/2013 Dept. of Physics Final Exam. Time: 2 hrs.

Q1) You are given the operator U  eiap , with a is a constant and p is the momentum operator. a) Show that U is unitary.

A  A 1 1 b) Using the identity e Be  B  A, B A,A, B A,A,A, B⋯, Find UxU † 2! 3!

Q2) Using L l,m   l(l 1)  m(m 1) l,m 1 L l,m   l(l 1)  m(m 1) l,m 1 m m a) Find LY1 , for all values of m. b) LY1 , for all values of m.

Q3) Consider a particle in a spherical potential well of radius a, i.e., 0 r  a V (r)    elesewhere U r Find the normalized wave function of the ground state energy (n=1) . By letting Rr  the radial r equation reduces to

2 d 2U ll 12  x 1   V r U r  EU r sin 2 ax dx   sin 2ax 2  2        2 dr  2r  2 4a

Q4) Consider a particle with the Hamiltonian

1 2 2 1 2 2 H  H  gxy with H  ( p  p )  k(x  y ) , o o 2 x y 2 is the Hamiltonian of the two-dimensional, isotropic harmonic oscillator. Using perturbation method, find the first order correction to the energy eigenvalue of the second excited state (n=2)

Q5) a) If S s,ms   s(s 1)  ms (ms 1) s,ms 1

S s,ms   s(s 1)  ms (ms 1) s,ms 1 find the matrix representation of S+ and S- b) Uing J  L S , Find L  S The IUG QUANTUM THEORY 18/1/2012 Dept. of Physics Final Exam. Time: 2 hrs.

Q1) Expressing p and x in terms of the annihilation and creation operators ( a and a† ), and using the

A  A 1 1 identity e Be  B  A,B A,A,B A,A,A,B ⋯ 2! 3! find the expectation value

0 eiqx p2 eiqx 0

where q is constant and p is one dimensional.

Q2) Consider a particle in a spherical potential well of radius a, i.e.,

0 r  a V (r)    elesewhere

U r Find the normalized wave function of the ground state energy (n=1) . By letting Rr  the radial r equation reduces to

2 d 2U ll 12   2   2 V rU r  EU r 2 dr  2r 

------  x 1 x  a  a† , p  i a†  a ,  sin 2 ax dx   sin 2ax 2 2 2 4a  1     1 2     L2  2  sin    L  Q3) Knowing that  2 2  z and  sin     sin    i  L2 2     T   r 2  , show that for a particle moving in a conservative force with 2 r 2 2 r 2 r  r  potential energy V

V a) H, L   i z  a) H, L2  V , L2  c) What would the above two results conform for central force problem?

Q4) Consider a particle with the Hamiltonian

H  Ho  gxy with 1 2 2 1 2 2 H  ( p  p )  k(x  y ) , o 2 x y 2 is the Hamiltonian of the two-dimensional, isotropic harmonic oscillator. Using perturbation method, find the first order correction to the energy eigenvalue of the second excited state (n=2)

------  x 1 x  a  a† , p  i a†  a ,  sin 2 ax dx   sin 2ax 2 2 2 4a The Islamic University of Gaza QUANTUM THEORY 4/9/2010 Department of Physics Final Exam. Time: 120 min.

Q1) Consider a particle of mass m confined in a one-dimensional box such that

0  a  r  a V (r)   otherwise

Prove that eigenfunctions and the eigen values are

1 n n  cos x, n is odd a 2a 1 n n  sin x, n is even a 2a n2h2 En  32ma2

Q2) Using the perturbation theory, find the eigenvalues of the modified Hamiltonian p2 1 H   m 2 x2 x 2m 2

Q3) Refering to the previous problem, solve the Hamiltonian exactly to find the wave functions and the corresponding iegen values.

Q4) Consider the H-atom in a uniform electric filed with the Hamiltonian p2 e2   H   eE  r , 2m r Using the perturbation method to find the eigen values of such a problem. You are given 1 1 2 2  1   r   r  1   r   r      1e 2ao   1    e 2ao cos 200  3   2a  210 2  3   a   2ao    o   2ao    o 

Q5) Use the variational method to find the minimum energy for the Hhamiltonian

p2 e2 H  2m r

 r Hint: the trial wavefunction is   e  with  is a variational parameter.