CHEM 442 Lecture 3 Problems
3-1. List the similarities and differences between the classical- and quantum-mechanical
equations of motion. How does the latter account for the quantization of energy and
wave-particle duality?
3-2. Suggest mathematical functions that represent a sinusoidal wave of amplitude A,
frequency n , and wavelength l .
3-3. Justify the mathematical form of the linear momentum operator, -i
d
, by applying
dx
it to the simple sinusoidal wave of 3-2.
3-4. Justify the mathematical form of the energy operator, i
d
, by applying it to the
dt
simple sinusoidal wave of 3-2.
3-5. Write down the time-independent and time-dependent Schrödinger equations of a
1
particle with mass m confined in the one-dimensional potential kx 2 , where k is some
2
constant.
3-6. What is an eigenvalue equation?
3-7. Starting with Hamilton’s equation of motion for classical mechanics,
obtain Newton’s equation of motion, F = ma .
p2
+V = E,
2m
d2
d
+ 2 + 3x̂ , on the following functions: (a) 2x , (b) eax , (c)
3-8. Act the operator,
2
dx
dx
.
sin kx
3-9. An operator Ŵ is said to be linear if Ŵ éë c1 f1 ( x ) + c2 f2 ( x ) ùû = c1Ŵf1 ( x ) + c2Ŵf2 ( x ) . In
quantum mechanics, all operators are linear. Determine whether the following operators
d2
d
3
x 2 , (d) ( ) , (e) ( ) . In the last two, ( )
are linear or nonlinear: (a)
, (b)
2 , (c)
dx
dx
is where the function acted on by the operator goes.