PHY4604 Fall 2007 Problem Set 4

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PHY 4604 Problem Set #4 Due Wednesday October 17, 2007 (in class) (Total Points = 110, Late homework = 50%)

Reading: Griffiths Chapter 3 and the Appendix.

Useful Integrals: π=∫+∞

∞−

dxx

x2

2 )(sin .

Problem 1 (25 points):. (a) (2 points) Show that the sum of two hermitian operators is hermitian (b) (2 points) Suppose that Hop is a hermitian operator, and α is a complex number. Under what condition (on α) is αHop hermitian? (c) (2 points) When is the product of two hermitian operators hermitian?

(d) (2 points) If dxdOop = , what is ↑

opO ?

(e) (2 points) Show that ↑↑↑ = opopopop ABBA )( . (f) (2 points) Prove that [AB,C] = A[B,C] + [A,C]B, where A, B, and C are operators.

(g) (2 points) Show that dxdfixfp opx h−=)](,)[( , for any function f(x).

(h) (2 points) Show that the anti-hermitian operator, Iop, has at most one real eigenvalue (Note: anti-hermitian means that opop II −=↑ ).

(i) (2 points) If Aop is an hermitian operator, show that 02 >≥< opA . (j) (2 points) The parity operator, Pop, is defined by PopΨ(x,t) = Ψ(-x,t). Prove that the parity operator is hermitian and show that 12 =opP , where 1 is the identity operator. Compute the eigenvalues of the parity operator. (k) (2 points) Idempotent operators have the property that opop PP =2 . Determine the eigenvalues of the idempotent operator Pop and characterize its eigenvectors. (l) (3 points) Unitary operators have the property that 1== ↑↑

opopopop UUUU , where 1 is the identity operator. Show that the eigenvalues of a unitary operator have modulus 1. Show that if <Ψ|Ψ> = 1 then <UΨ|UΨ> = 1 (i.e. unitary transformations preserve inner products). Prove that if Aop is hermitian then opiAe is unitary. (Hint: see Griffiths appendix) Problem 2 (15 points): Schrödinger’s equation says

ttxitxHop ∂

Ψ∂=Ψ

),(),( h ,

where the Hamiltonian operator is given by )(2/2 xVmpH xop += . (a) (2 points) Prove that Hop is hermitian.

PHY4604 Fall 2007 Problem Set 4

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(b) (5 points) Prove that

>∂∂

<+><>=<t

OOHiO

dtd op

opopop ],[h

.

If the operator Oop does not depend explicitly on time and if it commutes with the Hamiltonian, then <Oop> is constant in time (i.e. it is conserved). (c) (5 points) Prove that

dxdVxTpx

dtd

opxop −><>=< 2)( ,

where T is the kinetic energy operator (i.e. H = T + V). (d) (3 points) Use (c) to show that in a stationary state

dxdVxT >=<2 .

This is called the virial theorem. (Hint: see Griffiths section 3.5.3 and problem 3.31) Problem 3 (30 points): Consider an infinite square well defined by V(x) = 0 for 0 < x < L, and V(x) = ∞ otherwise. The stationary state position-space wave functions are given by

h/)(),( tiEnn

nextx −=Ψ ψ with )/sin(2)( LxnL

xn πψ =

and the eigenvalues are 2

222

2mLnEn

hπ= , and n is a positive integer.

(a) (10 points) Find the momentum-space wave function ),( tpxnΦ for the nth stationary state. (b) (5 points) Graph 2

11 |),(|),( tptp xx Φ=ρ as a function of px for the ground state (i.e. n = 1). (c) (5 points) Graph 2

22 |),(|),( tptp xx Φ=ρ as a function of px for the 1st excited state (i.e. n = 2). (d) (10 points) Use the momentum-space wave function ),( tpxnΦ to calculate the expectation values of px and px

2 for the nth stationary state and compare your answer with your answer to problem set #2 problem 2(c). (Hint: see Griffiths section 3.4 and problem 3.28) Problem 4 (20 points): Consider a three-dimensional vector space spanned by an orthonormal basis |1>, |2>, |3>. “Ket” vectors |α> and |β> are given by

|α> = i|1> -2|2> -i|3> and |β> = i|1> + 2|3>. (a) (5 points) Construct the “bra” vectors <α| and <β| in terms of the dual basis <1|, <2|, <3|. (b) (5 points) Find <α|β> and <β|α> and confirm that <β|α> = <α|β>*. (c) (10 points) Construct all nine matrix elements of the operator Aop = |α><β|, in this basis, and construct the matrix A. Is it hermitian? (Hint: see Griffiths section 3.6 and problem 3.22)

PHY4604 Fall 2007 Problem Set 4

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Problem 5 (20 points): The Hamiltonian for a certain three-level system is represented by the matrix

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

abc

baH

000

0

where a, b, and c are real numbers. (a) (5 points) What are the energy eigenvalues of this system (i.e. the allowed energies)? (b) (5 points) What are the (normalized) eigenvectors of H? (c) (5 points) If the system starts out at t = 0 in the state

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛>=

010

)0(| s

what is |s(t)> at later time t. (d) (5 points) If the system starts out at t = 0 in the state

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛>=

100

)0(| s

what is |s(t)> at later time t. (Hint: see Griffiths section 3.6 and problem 3.37)