EP2210, Principles of Quantum Mechanics, JulyNovember 2014

Exercise sheet 6

Essential mathematical formalism I

1. Eigen values and eigen functions of the momentum operator: Determine the eigen values and theeigen functions of the momentum operator. Establish the completeness of the momentum eigenfunctions.

2. The angular momentum operator: Consider the operator

L = i~d

d,

where is an angular variable. Is the operator hermitian? Determine its eigenfunctions and eigen-values.

Note: The operator L, for instance, could describe the conjugate momentum of a bead that isconstrained to move on a circle of a fixed radius.

3. Probabilities in momentum space: A particle of mass m is bound in the delta function well V (x) =a (x), where a > 0. What is the probability that a measurement of the particles momentumwould yield a value greater than p0 = ma/~?

4. The energy-time uncertainty principle: Consider a system that is described by the Hamiltonian

operator H.

(a) Given an operator, say, Q, establish the following relation:

dQ

dt=i

~[H, Q]+

Q

t

,

where the expectation values are evaluated in a specific state.

(b) When Q does not explicitly depend on time, using the generalized uncertainty principle, showthat

HQ ~

2

dQdt .

(c) Defining

t Q

|dQ/dt|,

establish that

Et ~

2,

and interpret this result.

5. Two-dimensional Hilbert space: Imagine a system in which there are only two linearly independentstates, viz.

|1 =

(10

)and |2 =

(01

).

The most general state would then be a normalized linear combination, i.e.

| = |1 + |2 =

(

),

Page 1

EP2210, Principles of Quantum Mechanics, JulyNovember 2014

with ||2+ ||2 = 1. The Hamiltonian of the system can, evidently, be expressed as a 22 hermitianmatrix. Suppose it has the following form:

H =

(a bb a

),

where a and b are real constants. If the system starts in the state |1 at an initial time, say, t = 0,determine the state of the system at a later time t.

Page 2