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 examp
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
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 ),
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
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
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 examp
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
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 const
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
clear
close all;
c=2.998e10;%cm/s
hbar=6.582e-16; %in eV*sec
m=5.11e5/c^2; %in eV/c^2
dx=1e-9; %0.1 Ang, in cm
tx=hbar^2/(2*m*dx^2);
title('Half-triangular well, \alpha=10^9 eV/cm: electron density di
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"
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
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
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 y
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 quantize
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
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
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 operat
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
c
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
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 proba
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 eac
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
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 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
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 molecu
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
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 d
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 wavefuncti