PHYS-4023 Introductory Quantum Mechanics
HW # 1
Chapters 1-3 [ Quantum Physics 3-rd Ed, Stephen Gasiorowicz ]
Instructor: Assistant Prof. Orion Ciftja
Name: ..................................................
Deadline: February.20.2004
Problem 1: Show that for a particle in a one-dimensional (1D) box, the mean-square
deviation of the particle from the center of the box is
D
E
(x − hxi)2 =
a2
6
1− 2 2
12
nπ
where a is the length of the 1D box and n = 1, 2, 3, . . .
Problem 2: Consider a particle in a one-dimensional (1D) box of length a. Calculate
∆x =
q
h(x − hxi)2 i =
q
hx2 i − (hxi)2 and ∆p =
q
h(p̂ − hp̂i)2 i =
q
hp̂2 i − (hp̂i)2 . Calculate
∆x ∆p and verify whether the Heisenberg uncertainty principle is satisfied. (p̂ is the linear
momentum operator for the x direction).
Problem 3: A particle is in a one-dimensional (1D) box with sides at x = ±a/2.
Solve the stationary Schrodinger equation and find the allowed energy eigenvalues En and
corresponding normalized eigenfunctions, φn (x). Are the energy eigenvalues
En =
h̄2 π 2 n2
; n = 1, 2, . . .
2ma2
as in the case of the 1D box with sides at x = 0 and x = a ?! Are the normalized
eigenfunctions always
s
φn (x) =
2
nπx
sin
a
a
; n = 1, 2, . . .
as in the case of the 1D box with sides at x = 0 and x = a ?!
Problem 4: Consider an electron in a macroscopic one-dimensional (1D) box of size
a = 2 cm.
(a) What value n corresponds to an energy of 1.5 eV ?
(b) What is the difference in energy between the state n and n + 1 in that energy region ?
Problem 5: A particle is in the ground state of a one-dimensional (1D) box with sides
at x = 0 and x = a. Very suddenly the right side of the box moves to x = b = 2 a. What
is the probability that the particle will be found in the ground state for the new potential
(larger box) ?! What is the probability that it will be found in the first excited state of the
new potential (larger box) ?!