Homework 1 Problem 1 (4 points) Consider an electron on a linear triatomic molecule formed by three equidistant atoms. We use  A ,  B , C to denote three orthonormal states of this electron, corresponding respectively to three wave functions localized about the nuclei of atoms A, B, C . We shall confine ourselves to the subspace of the state space spanned by  A ,  B and C . When we neglect the possibility of the electron jumping from one nucleus to another, its energy is described by the Hamiltonian H 0 whose eigenstates are the three states  A ,  B ,  C with the same eigenvalue E0 . The coupling between the states  A ,  B ,  C is described by an additional Hamiltonian W defined by: W  A  a  B
A
W  B  a  A  a C A
A
W  C  a  B
where a is a real positive constant. Calculate the energies and stationary states of the Hamiltonian H  H 0  W . The electron at time t  0 is in the state  A . Discuss qualitatively the localization of the electron at subsequent times t . Are there any values of t for which it is perfectly localized about atom A, B or C ? Let D be the observable whose eigenstates are  A ,  B ,  C with respective eigenvalues  d , 0, d . D is measured at time t ; what values can be found, and with what probabilities? Give a physical interpretation of D . Problem 2 (4 points) (1) A non‐relativistic particle ( E 
p2
)is confined in a in a two‐dimensional rectangular well with sides 2m
LX  LY and infinitely high walls. The particle is non‐relativistic and has mass m. Find the eigenvalues and eigenstates (including normalization coefficients) of the particle. Find the density of states for two cases (i) standing waves, rigid boundary condition, (ii) running waves, periodic boundary condition. (you must get the same results in both cases) . See lecture notes for 3d case. (2) Now consider ultra‐relativistic particle ( E  cp ) confined in the same potential well. Find eigenstates and eigenvalues for this problem. Find the density of states for either of the boundary conditions mentioned above. ( This part of the problem can be applicable to describe a state of an electron in graphene – two dimensional carbon sheet with honeycomb atomic structure, Electrons in this two‐
dimensional material have “relativistic” dispersion relation, E  vp ) Problem 3 (2 points) A particle of mass m is confined in a one ‐ dimensional harmonic potential with potential energy given 1
by V ( x)  m 2 x 2 . A wave function of the ground state in this potential has a Gaussian form
2
1/4
m
 m 
2
 ( x)  A exp( x ) , where  
and the normalization constant A  
 . 2
  
(1) Find x 2 , an expectation value of a squared displacement of the particle. (2) Compute the wave function of the particle in the momentum representation, A( p ) . Here you will need to evaluate integral with a complex exponent. Deal with is as if it is a real number and you will be fine. Some integrals needed for this problem are listed below. Not all of them are actually needed Gaussian integrals 
1 
 exp( x )dx  2
2

0

 x exp( x
2
)dx 
0
1
2

1 
0 x exp( x )dx  4  3
2

x
0
3
2
exp( x 2 )dx 
1
2 2