1. Complex Numbers
While the values of physical observables must be real numbers, quantum mechanical amplitudes are, in general, complex numbers. Consequently, some facility with the properties of
complex numbers is neede
Redo Exam 2
dx xn1 cos(ax)
dx xn cos(ax) =
dx xn1 sin(ax)
xn ax n
dx xn e ax =
dx xn1 e ax
dx xn sin(ax) =
dx xn ex = n!
A complex number has
Solutions Exam 2
1. (a) (2 pt) The eigenvalues of H are given by
det(H I ) = det
= (2E ) (2E )2 E 2 ) = 0 = E, 2E, 3E.
(b) (3 pt) If e1 corresponds to E , e2 to 2E and e3 to 3E , eigenvectors are
Solutions Exam 1
1. (a) (2 pt) The term Aeikx describes an incoming wave travelling from x = toward
x = 0, Beikx is the reected wave travelling toward and Ceix represents the
transmitted wave travelling from x = 0 toward x = +.
dx xn eax =
dx x2n ea
A complex number has the form z = x + iy , where x and y are real. It satises the usual rules of
algebra with i2 = 1, so
Problem Set 1
1. Math Review Section 1 Exercise 3 and 5.
2. Math Review Section 2 Exercise 3.
3. Math Review Section 3 Exercise 1.
4. Griths Problem 1.11.
5. Griths Problem 1.12.
6. The wave function for a particle is given by
Problem Set 2
7. Griths 1.4.
8. Griths 1.5.
9. Griths 1.9 (a), (c), (d).
10. Determine x and p for the state
where a and k are real.
11. Griths 2.4.
2 /2a2 +ikx
Problem Set 3
12. The particle in Example 2.2 of Griths has the initial wave function
(x, 0) =
x(a x) .
(a) Graph (x, 0). Which stationary state does it most closely resemble? On that basis,
estimate the expectation value of th
Problem Set 4
16. A particle of mass m in an innite potential well of width a has the initial wave function
(x, 0) = A sin3 (x/a) .
(a) Normalize (x, 0).
(b) Express (x, 0) in terms of the energy eigenstates
n (x) =
Problem Set 5
19. In class we solved for the bound states of the nite square well potential by analyzing the
even solutions with (x) = (x). Choosing 2mV0 a2 /h2 = 64, we found the lowest energy
to be E1 = 0.97V0 and the next energy t
Problem Set 6
23. A particle of mass m moves in the harmonic oscillator potential V (x) = 2 m 2 x2 .
(a) Show that the motion classical motion is restricted to the region 2E/m 2 x
2E/m 2 .
(b) Using the harmonic oscillator wave fu
Problem Set 7
28. In the attractive -function potential problem, let m/h2 = 1/a, which means that the
bound state eigenfunction and energy are
(x) = e|x|/a ,
If a particle moving in this potential is in the initial
Problem Set 8
31. Griths Problem A.9
32. Griths Problem A.23
33. Griths Problem A.25
34. Griths Problem A.28 Part (a)
35. An inner product for functions dened on 0 x can be dened as
g |f =
dx g (x)f (x)ex .
(a) Show that g |f is a
Problem Set 9
36. A linear transformation T is dened by the matrix
(a) Find the eigenvalues and normalized eigenvectors of T .
(b) Construct the similarity transformation S which diagonalizes T and show that ST S 1
Problem Set 10
41. The eigenstates of momentum are
p (x) =
dx (x) p (x) = (p p ) .
A quantum mechanical system is in the initial state (x, 0) = A/(x2 + a2 ).
(a) Normalize (x, 0).
(b) Expand (x, 0) in terms of th
Problem Set 11
45. Imagine a two-dimensional rectangular innite well in the x y plane with dimensions
(a) What are the energy eigenvalues and eigenfunctions for a particle of mass m moving
in this well?
(b) What are the rst ve
Problem Set 12
50. Solve the radial equation for a spherical well of depth V0 and radius a
V (r) =
with E < 0 and
V0 0 r a
(a) Show that the bound state energies are determined by the equation
n a cot(n
Extra Credit Problem Set 4
We proved that the eigenfunctions of the Hamiltonian corresponding to dierent eigenvalues
are orthogonal. To illustrate the power of this result, show by direct calculation that the
bound state bound functi
Solutions Problem Set 5
19. For the odd solutions, we have
for < x a
A sin x for a x a
2m(V0 + E )
Imposing the boundary conditions at x = a gives (5 pt)
a cot a = ka =
2 a 2 ,
Solutions Problem Set 9
36. (a) (4 pt) The eigenvalues of T are obtained from
det(T I ) = det
= 2 1 = 1 .
The eigenvectors are
(b) (6 pt) The inverse of the similarity transformation S is obtained by using the
Solutions Problem Set 12
50. (a) (5 pt) The solution to the Schrdinger equation for the spherical well is
AI sin(r) + BI cos(r) 0 r a
AII ekr + BII ekr
with = 2m(V0 + E )/h2 and k = 2mE/h2 . As r 0, u(r) r and as r ,
PHY 481 HW #3
Due: Thursday 10/ 11/ 12 in BPS 1420 before class
NOTE: You might want to make a xerox copy of your solutions before
you hand them in since this material will be covered on the rst midterm
(scheduled for thursday 10/18).
1. Two large metal p