Download INTRODUCTION TO QUANTUM MECHANICS I I mention in class

INTRODUCTION TO QUANTUM MECHANICS I
PROBLEM SET 4
due October 4th, before class
I.
DEGENERACY IN ONE DIMENSION, TUNNELING AND ALL THAT
I mention in class how the energy eigenvalues in one dimension are never degenerate. That may seem counterintuitive. In fact, consider two identical potential wells separated by a vast distance. One could imagine that a wave
function concentrated around one well would be degenerate in energy with an identical function concentrated around
the other well. To understand this better, let us consider two attractive δ function potentials separated by a distance
L:
L
L
V (x) = −λ δ(x − ) + δ(x + ) ,
(1)
2
2
with λ > 0.
1. Find the bound states and their energies.
2. Consider the limit L → ∞. How can we (approximately) write the eigenstates in terms of linear combinations of
the eigenstates of each well separately. Is the intuition described below (namely, that the corrected eigenstates
of the double well are essentially the eigenstates of the single well) correct?
3. We can abstract the double well potential into a more general situation. Imagine that initially we have a
hamiltonian that is degenerate
E0
0
Ĥ0 =
.
(2)
0
E0
These two degenerate states could correspond to the ground states of two identical single wells. Suppose now
we turn a very small interaction (maybe due to tunneling between the two wells)
0 T
.
(3)
ĤI =
T 0
Show that the eigenvalues of Ĥ = Ĥ0 + ĤI are no longer degenerate. Find also the eigenstates of Ĥ.
II.
HOW THE PREVIOUS PROBLEM EXPLAINS COVALENT BINDING OR “CHEMISTRY AS A
FOOTNOTE OF QUANTUM MECHANICS”
Let us make a crude model of a + H2 molecule (that is two protons and one electron). Imagine the protons are held
at fixed positions. The electron can be bound to each one of the protons, so we approximate the Hilbert space by a
two-dimensional space. When the electron is bound to one proton it has an energy E0 < 0. But there is always a
small probability for it to tunnel through the other proton (with a probability amplitude T ) so the full hamiltonian
(well, the crude approximation to it in the two-dimensional Hilbert space) is like the two-dimensional Ĥ above. Using
the result above you now what the energy of the electron will be.
1. Assuming that T should be larger when the protons are closer, make a qualitative drawing of the two energy
levels as a function of the distance between the protons.
2. In addition to the energy of the electron, there is a repulsive Coulomb energy between the protons. Make a
qualitative plot of the sum of Coulomb plus electron energy. The minimum is the preferred distance between
the two protons, that is, the size of the + H2 molecule.
That is the quantum mechanical explanation of the fact that a “shared electron” binds atoms together, that
always baffled me in high school. A better description of covalent binding that I can give here but along similar
lines is given in “Atomic Physics” by Max Born.
2
III.
PARTICLE IN A MAGNETIC FIELD
~ Its lagrangian is given by
Consider a charged particle moving in three dimensions subject to a magnetic field B.
L=
m~x˙ 2
~
+ q~x˙ .A,
2
(4)
~ = ∇ × A.
~ The goal of this problem is to apply the canonical quantization method and find a formulation of
with B
the quantum theory of a charged, spinless particle moving on an external magnetic field.
1. Verify that the equations of motion following this lagrangian yield Newton+Lorentz force.
2. Find the (classical) momentum canonically related to the position ~x.
3. What is the quantum mechanical operator associated with the canonical momentum above ?
4. What is the quantum mechanical operator for the velocity ~x˙ ?
5. Find the classical hamiltonian H.
6. find the quantum hamiltonian Ĥ.
7. Write the quantum hamiltonian operator in the position eigenbasis.
IV.
SQUARE WELL
Does the potential
V (x) =
−V0 , −L/2 < x < L/2
0, x > L/2 or x < −L/2
(5)
have a bound state for any l and V0 ? What is the binding energy? If you have any conceptual problem solving this
problem stop everything and solve all problems in Griffiths now.
V.
ONE DIMENSIONAL ATTRACTIVE POTENTIALS HAVE A BOUND STATE
1. Show that if two hamiltonians differ only in their potential and if V1 (x) < V2 (x) for all x, then the ground state
energy of Ĥ1 is smaller than the ground state energy of Ĥ2 .
2. Use this result and the square well problem to show that any attractive potential in 1D has at least one bound
state. This result is valid only in one dimension!
VI.
A FIRST LOOK AT THE VARIATIONAL PRINCIPLE
Show that the minimum of the functional
#
"
2
Z ∞
~2 dψ(x) 2
+ V (x)|ψ(x)|
[ψ] =
dx
2m dx −∞
(6)
under the constraint
Z
∞
dx |ψ(x)|2 = 1,
(7)
−∞
is given by the energy eigenstates. Then show that the minimum of the functional is given by the ground state.
Hint: you may want to brush up on the concept of Lagrange multipliers and the Euler-Lagrange equations of calculus
of variations.