3‒1. Evaluate fAg ˆ= , where A and f are given below:

A f

(a) SQRT 4x

(b) 3

3

3

xdx

d + axe−

(c) ∫1

0dx 323 +− xx

(d)2

2

2

2

2

2

zyx ∂∂+

∂∂+

∂∂ 423 zyx

3‒3. In each case, show that f(x) is an eigenfunction of the operator given. Find the

eigenvalue.

A f

(a)2

2

dx

dxωcos

(b)dt

dtie ω

(c) 322

2

++dx

d

dx

d xeα

(d)y∂

∂yex 62

3‒5.Write out the operator 2A for =A

(a)2

2

dx

d(b) x

dx

d + (c) 122

2

+−dx

dx

dx

d

Hint: Be sure to include f(x) before carrying out the operations.

3‒9. In Section 3.5, we applied the equations for a particle in a box to theπ electrons

in butadiene. This simple model is called the free‒electron model. Using the same

argument, show that the length of hexatriene can be estimated to be 867 pm. Show

that the first electronic transition is predicted to occur at 2.8×104 cm‒1. (Remember

that hexatriene has sixπ electrons.)

3‒10. Prove that if ψ(x) is a solution to the Schrödinger equation, then any constant

times ψ(x) is also a solution.

3‒11. In this problem, we will prove that the form of the Schrödinger equation

imposes the condition that the first derivative of a wave function be continuous. The

Schrödinger equation is

0)()]([2

22

2

=−+ xxVEm

dx

d ψψℏ

If we both integrate both sides from a ‒ ε to a + ε, where a is an arbitrary value of x

and ε is infinitesimally small, then we have

0)(])([2

2=−=− ∫

+

−−=+=

dxxExVm

dx

d

dx

d a

aaxax

ε

εεε

ψψψℏ

Now show that dψ/dx is continuous if V(x) is continuous.

Suppose now that V(x) is not continuous at x = a, as in

Show that

εψψψεε

)()2(2

2aEVV

m

dx

d

dx

drl

axax

−+=−−=+= ℏ

so that dψ/dx is continuous even if V(x) has a finite discontinuity. What if V(x) has an

infinite discontinuity, as in the problem of a particle in a box? Are the first derivatives

of the wave functions continuous at the boundaries of the box?

3‒15. Show that

2

ax =⟩⟨

for all the states of a particle in a box. Is there really physically reasonable?

3‒17. Show that

2122 )( ⟩⟨−⟩⟨= xxxσ

for a particle in a box less than a, the width of the box, for any value of n. If σx is the

uncertainty in the position of the particle, could σx even be larger than a?

3‒18. A classical particle in a box has an equi‒likelihood of being found anywhere

within the region 0 ≤ x ≤ a. Consequently, its probability distribution is

axa

dxdxxp ≤≤= 0)(

Show that 2ax =⟩⟨ and 322 ax =⟩⟨ for this system. Now show that ⟩⟨ 2x (Equation

3.32) and σx (Equation 3.33) for a quantum‒mechanical particle in a box take on the

classical value as ∞→n . This result is an example of the correspondence principle.

Equation 3.3222

222

23 πnaa

x −=⟩⟨ Equation 3.33

2122

232

−= n

n

ax

ππ

σ

3‒22. Prove that the set of functions

⋯,2,1,0)2()( 21 ±±== − neax anxi

n

πψ

is orthonormal over the integral ‒a ≤ x ≤ a. A compact way to express orthonormality

in the ψn is to write

mn

a

anm dxxx δψψ =∫−

∗ )()(

The symbol δmn is called a Kronecker delta and is defined by

≠=

=nm

nmmn

if0

if1δ

3‒27. Calculate ⟩⟨ p and ⟩⟨ 2p for the n = 2 state of a particle in a 1D box of length a.

Show that

a

hp =⟩⟨σ

3‒29. Discuss the degeneracies of the first few energy levels of a particle in a 3D box

when a ≠ b ≠ c.