t1 practice solution

8/12/2019 t1 Practice Solution

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33-234 Quantum Physics Spring 2010

Name: ___________________________________

Test 1 Solution

Put a box around the final formula in all cases and, in addition, around the

numerical answer where called for.

1. Suppose we have solved the time independent Schrodinger equation (for some specificpotential energy function) and obtained two wavefunctions, 1(x) and 2(x), with energyeigenvalues, E1 andE2, 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.

1(x, t) =1(x)ei1t

with 1 = E1/h(the deBroglie relation).

b. (10 points) If a particle is in the state described by 2(x), what is the probability offinding the particle within an interval,dx, of positionx? What are the units associated withthe probability function, P2(x) and with 2(x)?

P2(x)dx= |2(x)ei2t|2 dx= |2(x)|2 dx.P2(x)dxis dimensionless, therefore [P2(x)] = (length)

1 and [2(x)] = (length)1/2.

c. (8 points) If one forms a new function, (x, t), defined 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 thatthe factor of 1/

2 is necessary to maintain normalization thus, (x, t) is a normalized

wavefunction.

Only ifE1 =E2 will (x, t) be a solution of the time independent equation.(x, t) will always be a solution of the time dependent equation since it is a sumof solutions and the equation in linear.

d. (12 points) Write an expression, in terms of1,2and 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?

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P(x, t) = (x, t)(x, t) =1(x) e

i1t +2(x) ei2t

1(x) e

i1t +2(x) ei2t

(1)

= |1(x)|2 + |2(x)|2 +1(x)2(x)ei(12)t +ei(12)t

(2)

= |1(x)|2

+ |2(x)|2

+ 21(x)2(x)cos[(1 2)t] (3)so this probability function is a function of time and it oscillates with frequency1 2.

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 finite value or a non-zerovalue for E, the standard deviation of the energy?

< E >=

dx(x, t)E(x, t) =ih

dxt

ForE1=E2, E >0. IfE1 = E2, then the energy is a precisely defined constantof the motion and E= 0; further, < E >=E1= E2.

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

f(x, t) =Aei(kxt). (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: first writeA in polar form.

Re(z)

Im(z)

1 1

1

1 A

A= 1 +i=

2ei/4, so f(x, t) =

2 ei/4 ei(kxt) =

2 ei(kxt+/4).

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

|f(x, t)

|2 = 2.

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

L[f(x, t)] =A ei[k(x+x0)t] =eikx0f(x, t), so it is an eigenfunction with eigen-valueeikx0.


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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 ofks specifiedbyk, we calculated a trial wavefunction of the form

(x, t) =A ei(k0x0t) sin . (3)

Here,= 12

k(xvgt) withvg = ddkk0

being the group velocity. We assumed thatk


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g. (5 points) Is Eq. ?? a solution of the time-dependent Schrodinger equation for afree particle 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.

The function is a solution to the time dependent equation since it is composedof a sum of complex exponentials that are the origin (so to speak) of the timedependent equation.

The function cannot be an energy eigenfunction or a solution to the timeindependent equation since (1) the probability is a function of time, (2) thefunction cannot be factored into a product of a space dependent function and atime dependent function, (3) the function contains a range of values (and thatcorresponds to a range of energies), and (4) the function has a finite time width(at least in the sense that we defined the width of sin /) and therefore, by theUncertainty Principle, has to have a finite E. Any one of these explanationswould be sufficient.

t1 practice solution

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