t1 Practice

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33-234 Quantum Physics Spring 2010

Name: ___________________________________

Test 1

An equation sheet is on the last page.

Put a box around the nal formula in all cases and, in addition, around thenumerical answer where called for.

1. Suppose we have solved the time independent Schrodinger equation (for some specicpotential energy function) and obtained two wavefunctions, 1 (x) and 2 (x), with energyeigenvalues, E 1 and E 2 , respectively. Assume that 1 and 2 are real functions.

a. (5 points) What is the form of the full, time dependent wavefunction, 1 (x, t ), as-sociated with 1 (x)? Use a deBroglie relation to express 1 (x, t ) in terms of a frequency,1 .

b. (10 points) If a particle is in the state described by 2 (x), what is the probability of nding the particle within an interval, dx, of position x? What are the units associated withthe probability function, P 2 (x) and with 2 (x)?

1


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c. (8 points) If one forms a new function, ( x, t ), dened as

(x, t ) = 1 2 [1 (x, t ) + 2 (x, t )] ,

under what condition is ( x, t ) a solution of the time independent Schrodinger equation? Inthe general case, is ( x, t ) a solution of the time dependent Schrodinger equation? Note that the factor of 1/ 2 is necessary to maintain normalization thus, (x, t ) is a normalized wavefunction.

d. (12 points) Write an expression, in terms of 1 , 2 and the associated time dependentfunctions, for the probability function P (x, t ) associated with ( x, t ). You should write outthe expression in explicit form (not leaving it as |...|

2 ) and simplify any terms you can. Willthis probability be a function of time?

e. (5 points) Given the function ( x, t ), write an expression you could use to compute< E > , the expectation value for the energy. Would you expect a nite value or a non-zerovalue for E , the standard deviation of the energy?

2


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2

2. A complex function, f (x, t ), is written as

f (x, t ) = Ae i (kx t ) . (2)

a. (10 points) If A is the complex number represented in the following graph of the

complex plane, write f (x, t ) as a product of a single real number and a single complexexponential. Hint: rst write A in polar form.

Re(z)

Im(z)

1 1

1

1 A

b. (5 points) Evaluate |f (x, t )|2 for the value of A given in part (a).

c. (10 points) If L is the displacement operator dened as L [g(x)] = g(x + x0), whatis L [f (x, t )], where f (x, t ) is the function given above? Is f (x) an eigenfunction of thisoperator? If so, what is the eigenvalue. Show your work. You may use the form given by Equation 2 .


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3

3. To describe the wave-like propagation of a particle of mass, m , we want a wave-likefunction that moves through space with the speed of the particle. By adding together (thatis, by doing an integral of) harmonic traveling waves, ei (kx (k )t ], with a range of ks speciedby k, we calculated a trial wavefunction of the form

(x, t ) = A e i (k 0 x 0 t ) sin . (3)

Here, = 12 k (x vg t) with vg = d

dk k 0 being the group velocity. We assumed that k


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4

e. (5 points) Taking account of the denition of , does the position, xm , of the maximumvalue of sin depend to time, t? If so, at what speed does it move? Write this speed in termsof the dispersion relation, (k).

f. (5 points) If (k) = 0 cos(ka ), with 0 and a being constants, what is vg at k = k0?

g. (5 points) Is Eq. 3 a solution of the time-dependent Schrodinger equation for a freeparticle in a region of space with V (x) = 0? Is it a solution of the corresponding time-

independent Schrodinger equation? Justify your answer but you do not have to explicitlydemonstrate your answer through substitution.


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Test 1 Equations

= c

(free photons)

K = 12

mv 2 = p2

2m (non relativistic free masses)

(x, t ) = dk A(k) ei [kx (k )t ]x k = 4 (square packet)t = 4 (square packet)

x p

2

t E

2x = x

px = i x

K = p2

2m

E = i t

= 1 .054 10 34J s = 6 .58 10 16eV s

h = 6 .626 10 34J s = 4 .14 10 15eV se = 1 .602 10 19 C

m e = 9 .11 10 31 kg

m p m n = 1 .67 10 27 kg

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