Physics 401 - Homework #11
1) A spherical harmonic (three points). Apply the definition of the spherical harmonics
to calculate the explicit functional form for Y31( The definition is given, for example,
in Griffiths, equations 4.27, 4.28, and 4.32.
2) Lx
PHYS401 Quantum Physics I
- FINAL EXAM
No books, calculators, or notes
Spring 2012
Name: _
1. Consider an electron bound to a two-dimensional infinite quantum well with sides of length
and
.
a. Write down the time-independent differential wave equation go
Physics 401 - Homework #7
1) Orthogonality of the stationary states of the harmonic oscillator (three points).
Since energy is an observable quantity, it is represented by a Hermitian operator ( H ),
and this guarantees that the energy eigenstates (or sta
Physics 401 - Homework #11
1) An asymmetric finite potential well. Consider the asymmetric finite potential well shown
below:
a) (three points) Suppose that there exists a ground state with energy E0 as labeled in the figure.
Make a rough sketch of what y
Physics 401 - Homework #10
1) We define the quantum mechanical probability current vector J in three dimensions in
such a way that it satisfies a "continuity equation":
P(r , t )
J
0
t
where P(r , t ) is the probability density in three dimensions. (i.e
Physics 401 - Homework #12
1) A spherical harmonic (three points). Apply the definition of the spherical harmonics
to calculate the explicit functional form for Y31( The definition is given, for example,
in Griffiths, equations 4.27, 4.28, and 4.32.
2) Lx
Phys 401 Exam #2 Formula Sheet
1
2
e
ik ( x x )
dk ( x x ) ,
m n mn ,
n n I,
n
1
2
e
i ( k k ) x
dx (k k )
k k (k k ) ,
k k dk I ,
k (k ) ,
x ( x)
n an ,
x x ( x x )
x x dx I
an n ,
(k ) k dk ,
( x) x dx
x n n ( x) ,
xk
1
x x ( x x )
n
2
e ikx ,
C C
A
1. Using the Fourier Transform, find the solution f(x,t) of the drift-diffusion equation
subject to initial conditions f(x,t=0)=(x):
Where D (diffusion coefficient) and v (velocity) are positive constants. Interpret the
result.
2. Hermitian matrices:
a. W
PHYS401 HW1: due 3PM Monday Feb. 6
Complex Numbers:
1.Simplify this number: ii . Is it real/imaginary/complex?
2.Write this number in complex Cartesian form (real plus imaginary parts): -11/3. Derive a
general expression for fractional powers of -1 (i.e.
Physics 401 - Homework #9
1) The time evolution operator in matrix mechanics. In Homework #8, we showed that the
time-evolution operator takes an initial state wavefunction and "translates it in time" as described by
the Schroedinger equation. In this pro
Physics 401 - Homework #8
1) Three-state system. Suppose that we have a physical system for which there are only
three states. For example, perhaps the system is a molecule where the atoms can take on
three different arrangements. Regardless of how we int
Physics 401 - Homework #3
1) Expectation values in stationary states (three points). When the state of a quantum
mechanical system is an eigenfunction of the Hamiltonian, the expectation values of all
observables are constant in time. For this reason we r
Physics 401 - Homework #4
1) Incompatible observables (two points each). For this question, you do not need to do
any calculations, except a very brief one for part (c). For all others, just explain your
reasoning.
In quantum mechanics, if the measurement
Physics 401 - Homework #2
1) Expectation value of a discrete variable (six points total). The "quantum" in quantum
mechanics refers to the fact that the energy of bound states is discrete, or quantized, in
microscopic systems. Suppose that we have 15 iden
Physics 401 - Homework #1
1) (4 points total) Convert the following complex numbers into the z x iy and
z Ae i forms. State explicitly the values for x, y, A, and for each.
a)
ii
b) 1
1 i
2) (4 points total) Prove the following identities for complex numb
Physics 401 - Homework #5
1) A quantum mechanical state in two bases (12 points total). A quantum mechanical
particle-in-a-box is in a superposition of two stationary states, the n = 4 state and the n =
5 state. The superposition is an equal mixture of th
Physics 401 - Homework #6
1) Expectation value of a squared Hermitian operator (three points). In many
respects, Hermitian operators are analogous to a real numbers. For example, we say that
an operator is Hermitian if it is equal to its Hermitian conjuga
Physics 401 - Homework #7
1) Time evolution of expectation values (seven points total). If an observable (A) has a
quantum mechanical operator ( A ) which does not depend on time, then the time-rate
change of the expectation value of that observable in an
Phys 401 Final Exam Formula Sheet
i
d
H
dt
i
d
ci (t ) H ij c j (t ) , ci i t , H ij i H j
dt
j
A
t
ij
A ji
U , F UFU 1 , U mi m i , U 1 U t
d
E0 A
1
0 i dt c1 (t ) E 0 c1 (t ) Ac 2 (t )
H
A E , up 0 , down 1 ,
0
i d c 2 (t ) Ac1 (t ) E 0 c 2 (t
1. Implement the finite-differences approximation: write a MATLAB script to solve the problem of an
electron in a triangular well,
V(x)= x for x>0, infinite otherwise, where =109 eV/cm.
Plot the probability density nn for n=1,2,3 and the energy spectrum E
Physics 401 - Homework #1 - Due Wednesday September 9th
1) (4 points total) Convert the following complex numbers into the z x iy and
z Ae i forms. State explicitly the values for x, y, A, and for each.
i
a) i
b) 1
1 i
2) (4 points total) Prove the follow
Physics 401 - Homework #4 - Due Wednesday October 7th
1) Incompatible observables (one point each). For this question, you do not need to do
any calculations, except a very brief one for part (c). For all others, just explain your
reasoning.
In quantum me
Physics 401 - Homework #6 - Due Wednesday October 21th
1) Expectation value of a squared Hermitian operator (three points). In many
respects, Hermitian operators are analogous to a real numbers. For example, we say that
an operator is Hermitian if it is e
Physics 401 - Homework #11 - Due Wednesday December 2nd
1) An asymmetric finite potential well. Consider the asymmetric finite potential well shown
below:
a) (three points) Suppose that there exists a ground state with energy E0 as labeled in the figure.
Physics 401 - Homework #8 - Due Wednesday November 11th
1) Three-state system. Suppose that we have a physical system for which there are only
three states. For example, perhaps the system is a molecule where the atoms can take on
three different arrangem
Physics 401 - Homework #7 - Due Wednesday October 28th
1) Time evolution of expectation values (seven points total). If an observable (A) has a
quantum mechanical operator ( A ) which does not depend on time, then the time-rate
change of the expectation v
Physics 401 - Homework #12 - Due Wednesday December 9th
1) A spherical harmonic (three points). Apply the definition of the spherical harmonics
to calculate the explicit functional form for Y31( The definition is given in Griffiths,
equations 4.27, 4.28,
Physics 401 - Homework #9 - Due Wednesday November 18th
1) The time evolution operator in matrix mechanics. In Homework #8, we showed
that the time-evolution operator takes an initial state wavefunction and "translates it in
time" as described by the Schr