Problems

Problem:

(a) Radioactive decay can produce neutrinos of either of two varieties, called ne and nm.  There is considerable interest in the possibility that neutrinos have a small mass m1, m2. Suppose that ne,m are linear combinations of the mass eigenstates m1,2, thus

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A nuclear reactor emits neutrinos of type ne .  A distance L away a detector is able to record the passage of ne .  Assume that mc2<<E1,2, so that E=Pc+m2c3/2P.  Show that the intensity of ne at L relative to that at the source is

 .

(b) Suppose that Dm2c4=1eV2 and that a high energy accelerator is emitting neutrinos of energy 1 GeV.  What is the optimum distance to place the detector to observe the effect of oscillations?

(c) The accelerator energy is reduced such that neutrinos emerge with only 1/45 of the energy that they had previously.  How does the optimal distance change?
Solution:

(a) Equating mass and energy we assume that the mass eigenstates are the eigenstates of the Hamiltonian, which we assume does not explicitly depend on time.

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  is the evolution operator.  Therefore ,
.  A second operator, which we denote by A, has eigenstates and .  It is the operator which distinguishes between the two varieties of neutrinos.

What is the probability that at time t a measurement of the observable A yields a neutrino of type ne? 

.

Here we assume that and are normalized, and since they correspond to different eigenvalues of H they are orthogonal.

We have 

).

.  For t we have Therefore

.

(b) varies between 0 and 1.  To see oscillations there is no optimum distance.  One must map out the number of ne as a function of distance over several periods.

The first minimum in the number of ne occurs when   =1 for the first time, i.e. when or when

(c) 

Problem:

In recent years physicist have been questioning the usual assumption that neutrinos are massless.  If the neutrinos are endowed with a small mass, then oscillations may occur between the different types, (e. g. ne, nm) provided that the types have different masses.  The weak-interaction eigenstates ne, nm are expressed as combinations of the mass eigenstates n1, n2 , which propagate with different frequencies due to their mass differences.  (In this problem we ignore the third neutrino, nt .)  As a result, if one starts with a pure ne beam, the oscillations result in a subsequent admixture of ne and nm in the beam.  Consider the transformation which defines the admixture,

Suppose a neutrino of well defined momentum pn in this beam is born with a definite type f at time t=0.  Then at that time its wave function is 

   

With time this will evolve as 

  

Since neutrinos are relativistic, this means 

(a) Show that the probability of finding the neutrino to be type f’ at a distance x from its source, if the original type was f, is

b) What is lmm'?

(c) Imagine adding an apparatus which can measure indirectly the mass of a given neutrino when it is produced.  For example, the mass of the neutrino produced in  could be determined accurately measuring the four vectors of the pion and muon.  Show that the uncertainty principle will lead to an elimination of the neutrino oscillation in this case (i.e. knowing the mass of the neutrino produced prevents it from oscillating to another type).

(Hint: The mass must be determined to a precision which allows one to discriminate between different types, i.e. )
Solution:

(a)   .

.

.

,

where  .

(b)

(c) Let denote the uncertainty in the square of the mass and let   If we assume that the neutrinos have a well defined momentum then

.  The uncertainty in the phase of the cosine is then given by

 

We have Dphase is not much smaller than , i.e. the phase is washed out.

Problem:

For the infinite well shown, the wave function for a particle of mass m, at t=0, is

pot1.gif (2299 bytes)

(a) Is an eigenfunction of the Hamiltonian?

(b) Calculate <x>, <px>, and <H> at t=0.
Solution:

(a) is not an eigenfunction of the Hamiltonian.

(b)

. y is an eigenfunction of the infinite square well with V=0 for 0<x<a. For the infinite square well we have <H>=E=<T>.

Therefore 

 

.

.

Problem:

A particle of mass m is inside a one-dimensional infinite well with walls a distance L apart.  One of the walls is suddenly moved by a distance L so that the wall separation becomes 2L.  The wall moves so suddenly that the particle wave function has no time to change during the motion.  Suppose that the particle is originally in the ground state.

(a) What is its energy E0 and wave function y0 before the width is doubled?

(b) What are the energy eigenvalues after the width is doubled?

(c) If we measure the energy after the width is doubled, what is the probability that it will not have changed?

(d) If we measure the energy after the width is doubled, what is the probability that the particle will have lost some energy?

(e) What is the expectation value of the energy before and after the doubling of the width?
Solution:

For a particle in an infinite well we have .

(a) .

(b) .

(c) For the energy to not have changed the particle must be found in the first excited state of the new potential (n=2)

(We integrate only from 0 to L since for x>L.)

(d) If we find the particle in the state n=1 of the new potential then the particle has lost energy.  (If we do not find it in n=1 then it has not lost energy.)

.

(e)    before and after the doubling of the width.

Problem:

Assume the operator A commutes with the Hamiltonian H of a conservative physical system.  Prove that in any state |y(t)> the probability of observing the eigenvalue a0 is independent of time.

Solution:

If A commutes with H we can find a common eigenbasis of A and H.  Let { |Eni,amj> } denote a common orthonormal eigenbasis of A and H for the state space E.

H |Eni,amj>=En |Eni,amj>, A |Eni,amj>=am |Eni,amj>.

Here i and j the degeneracy of the eigenvalues.

|y(0)> may be expanded in terms of the basis vectors.

.

Then

.

The probability of observing the eigenvalue a0 at time t is given by

, which is independent of time.

Problem:

Consider a one-dimensional problem.  Let the translation operator T(a) describe the operation T(a)y(x)=y(x+a), where a is a constant displacement.

(a) Show that this operator commutes with the Hamiltonian , if the potential has the periodic property V(x)=V(x+a).

(b) Let y(x) be an eigenstate of T(a) with eigenvalue c.  Show that c is a constant of motion.

Solution:

(a)   (Taylor series expansion).

.

Image2238a.gif (1527 bytes).

For any y(x) we have

  since V(x+a)=V(x). 

Thus [T(a),H]=0.

(b) T(a)y(x)=cy(x).  Since T(a) does not explicitly depend on time we have

 

Since y(x) is an eigenstate of T(a) we have <T(a)>=c.

Therefore c is a constant of motion.

y(x) represents the projection of |y> onto the basis vector |x>.  

What does T(a) do to the basis vectors {|x>} ?

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