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PHY 4604 Exam 1 Name:__________________________________

Monday October 3, 2005 (Total Points = 100)

Problem 1 (10 points): Circle true or false for following (1 point each).

(a) (True or False) One of the breakthroughs that lead to quantum mechanics was the idea of

associating differential operators with the dynamical variables.

(b) (True or False) The wave function (x,t) must vanish in a region of infinite potential.(c) (True or False) It is possible for a free particle to have a definite energy.

(d) (True or False) In quantum mechanics particles can enter the classically forbidden region

where V0 > E (i.e. KE < 0).

(e) (True or False) The operator(Aop-A

op) is hermitian.

(f) (True or False) IfAop and Bop are hermitian then Aop-Bop is also hermitian.

(g) (True or False) IfPop is the parity operator, Pop(x) = (-x), then Pop2 = 1.(h) (True or False) Solutions of Schrdingers equation of the form )()(),( txtx =

correspond to states with definite energy E.

(i) (True or False) Solutions of Schrdingers equation of the form )()(),( txtx = correspond to states in which the probability density

2|),(|),( txtx = is independent of time.

(j) (True or False) Schrdingers equation is valid for all velocities even when v c.

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Problem 2 Name:_________________________________________

Problem 2 (30 points): Consider an infinite square well defined by

V(x) = 0 forL/2 < x < +L/2,

V(x) = + otherwise.

We look for stationary states of the form h/)(),( tiEnn nextx = . Parity is agood quantum number in this problem since V(-x) = V(x) and hence the

stationary state solutions are either even or odd under parity as follows:

)()()( )()()( xxxPnnn

++++++ ==

)()()( )()()( xxxPnnn

==

(a) (5 points). Calculate the (normalized) even parity wave functions )()( xn

++ and their

corresponding energies, +nE .

(b) (5 points). Calculate the (normalized) odd parity wave functions )()( xn

and their

corresponding energies, nE .

(c) (3 points). What is the state of lowest energy (i.e. ground state) and what is its energy, E0. Isit a parity even or parity odd state?

(d) (3 points). What is the state with the 2nd

lowest energy (i.e. 1st

excited state) and what is its

energy, E1. Is it a parity even or parity odd state?

(e) (14 points). Suppose that a particle in this infinite square well has the initial

wave function at t = 0 given by

=A

x0

)0,(4/4/

4/||

LxL

Lx

>

where A is a constant. Determine the normalization A. If a measurement of the

energy of this state is made at a later time t, what is the probability that the measurement willyield the ground state energy, E0? What is the probability that it will yield the 1

stexcited state

energy, E1?

V = +infV = +infinity

Infinite Square Well

-L/2 +L/2

V = +infV = +infinity

Infinite Square Well

-L/2 +L/2

(x,0)

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Problem 2 Name:_________________________________________Scratch Paper

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Problem 3 Name:_________________________________________

Problem 3 (30 points): Suppose that particles with energy E > V0 enter from the

left and travel to the right and encounter both a delta-function potential and a

step-function potential at x = 0 as follows:

+=

)(

0)(

0 xVxV

0

0