Answers to drylab-II-The postulates - Cobalt

CHEMISTRY 373 - DRY LAB II
PRACTICE FOR THE QUANTUM MECHANICAL POSTULATES
Practice Problems for January
A. Questions on Postulate 1.
1. Which of the following mathematical functions is a well-behaved wave function:
(a) e kx2 ,K < x < +K, k > 0, (b) e kx2 ,K < x < +K, k > 0
(c) e imx ,K < x < +K, m a real number
(Answer: (a) This is a well behaved function. It is continuous, single-valued, and vanishes as
x ¸ ±K. (b) Blows up as x ¸ ±K. (c) This function is continuous and single-valued but is not
defined in the limit asx ¸ ±K.Þ
2. Show that the wave functions
1
f 1 ÝxÞ = 2 2 sin ^x
L L
1
, f 2 ÝxÞ = 2 2 sin 2^x
L L
for the first two levels of a particle in a 1-dimensional box are orthonormal.
B. Questions on Postulate 2 and commutators.
1. Find the quantum mechanical operators corresponding to the following classical quantities:
xp x , p 2 x + x 2 , p 2 x + p 2 y + g 2 Ýx 2 + y 2 Þ + Ux.
(Answers:
xp x ¸ åx å p x + å å
p xx = i¥ x
/
+ / x , Tricky. Later we will learn that xp /x /x x is a classical
observable and must correspond to a Hermitian operator in quantum mechanics. The operator
åå xp x is not Hermitian (next question) but å x å p x + å å
p xx is.
p x 2 + x 2 ¸ åp
x 2 + å x 2 = ¥ 2 /2
/x 2 + x 2 ,
p x 2 + p y 2 + g 2 Ýx 2 + y 2 Þ + Ux ¸ åp
x 2 + å p y 2 + g 2 Ý å x 2 + å y 2 Þ + U å x
= ¥ 2 /2
/x 2 + /2
/y 2 + g 2 Ýx 2 + y 2 Þ + Ux.
)
2. Which of the following operators are self-adjoint or Hermitian:
å p 2 z ,
åå xp y å y å p x ,
åå xp x
3. What is the adjoint of the operator å x å p x
(Answer:
å å p xx + i¥.
)
4. Evaluate the following commutator brackets:
where the Hamiltonian, H = ¥2
2m
ß å x, å p x 2 à, ß å x å p y å y å p x , å y å p z å z å p y à, ßH, å xà.
/ 2
/x 2 + VÝxÞ, is for a 1-dimensional system only.
(Answers: Use the fact that ß å x, å p x à = ß å y, å p y à = ß å z, å p z à = i¥ and ß å x, å p y à = 0, etc., as well as the
commutator identities ßAB,Cà = AßB,Cà + ßA,CàB andßA,BCà = BßA,Cà + ßA,BàC.
ß å x, å p x 2 à = 2i¥ å p x ,ß å x å p y å y å p x , å y å p z å z å p y à = i¥Ý å z å p x å x å p z Þ,
ßH, å xà = ß px 2
2m + VÝå xÞ, å xà = 1
2m ßå x, å p x 2 à = i m ¥ å p x
,

)
C. Questions on Postulate 3.
1. Which of the following functions are eigenfunctions of d2
dx 2
2. Solve the following eigenvalue equation:
(Answer:
iJ
Af = Jf ö xf x = f ö f = Nx iJ ¥ .
¥
)
3. Show that the two operators
A = x / /x
e kx2 ,x 2 ,coskx + sinkx.
Af = Jf, A = å x å p x .
/2
+ K, B = x2 + ÝJ KÞ
2
/x
and identify the eigenvalues:
(with J,K constant) commute. Suppose that fÝxÞ is an eigenfunction of A with eigenvalue J, i.e.,
Af = Jf.
Then, find f. Also, show that f is simultaneously an eigenfunction of the operator B with
eigenvalue ÝJ KÞ 2 .
(Answer:
fÝxÞ = Nx JK , then compute Bf.
)
D. Questions on Postulate 5.
1. For a particle in a 1-dimensional box, calculate the expectation values < å x å p x > and < å p x
å x > for
the ground state. Are they real Are they equal Explain. Are the expectation values the same
for all states of the particle in a 1-dimensional box
(Answer:
i¥ 2 L
X
L sin
n^x d
x sin n^x
0 L dx L
dx = 1 2 i¥ 2n^cos2 n^n^cosn^sinn^
n^ = 1 2 i¥
i¥ 2 L
X
L sin
n^x d
x sin n^x dx = 1 i¥ cosn^sinn^3n^+2n^cos2 n^
0 L dx L
2 n^ = 1 i¥ 2
The operators for å x å p x and for å å
p xx are not self-adjoint.)