Particle in a box

Boundary value problem where one needs to find the solution for the problem given that the potential is ∞ at the boundaries. Thus the particle is completely confined to a region and the wave function that describes this state must reflect that condition.

Let us ask the question what is an allowable solution or a physical state that makes sense in this problem.

ANSWER: A particle in the box but not on the boundary.

Not normalized but any of the above functions is OK.

Method divide the space up into regions and solve the problem in each region:

Region 1: x<0

Region 3: x>0

Region 2: a>x>0 Solutions that are energy eigenstates.

Examine the Hamiltonian and find the eigenstates

To get the BC one has B=0 and [C_n=-D_n=1/(2i)]

To normalize the state

We find that :

· Energy eigenstates are not eigenstates of either momentum or position.

· We can find the average location and average momentum for any eigenstates.

· There are many acceptable states that have a definite energy.

· The time evolution for a specific state is a solution to the Schrodinger equation

· A general solution, a legitimate possibility for the particle located in a box is

A solution at some time that will now need to evolve based on the Schrödinger equation.

Be clear:

· is a solution or a prepared state at a fixed time that must evolve

· is a very specific solution it usually has the property that the wavefunction is a solution to some operator eigenequation and therefore there are an associated set of eginevalues labeled by n.

· implies that that time dependence of this state is included. This can be very general. So far we have examined the particularly simple time evolution of an energy eigenstates.

· Constructed the most general state by combing eigenstate.

Using separation of variables one can get a wave equation solution for waves on a string with fixed end points. The solutions for the separated spatial part are the same as above. For fixed points at 0 and a, the solution will be similar to the above. The relation between the wave number k and is different. For waves the ratio is fixed. For quantum states the ratio depends on k. This means that the propagation of the eigenstates varies with k. One interesting property of waves is that they propagate without distortion. This is usually referred to as non-dispersive.

Imagine a string. Distort the string into an initial disturbance. This can be described in terms of f(x) the displacement from equilibrium as a function of the position along the string. If one examines the system later the string will have a disturbance that travels to the left and to the right. These disturbances keep their original form

f(x,t)= A_of(x-vt) + B_of(x+vt)

Consider the earth in orbit around the sun.

State the earth is located at some distance from the sun and moving with some momentum.

Result is an orbit.

State the earth is located at some distance from the sun and moving with some momentum.	prepared quantum state with specified (all possible) information using wave function.
This state evolves in time based on · Newton’s laws · Lagrangian · Hamiltonian · Your favorite formulation	This state evolves in time based on · Newton’s laws · Lagrangian · Hamiltonian · Your favorite formulation
Classical formulation	Quantum formulation
Result is an orbit.

In referring to QS a very interesting property is that the overall phase can not be measured. There can be no measurable difference between

is a real number.

This is an important property but one must be extremely careful in applying this rule. The key is the word overall. Modifying any part of a quantum state by changing phase factors will result in a possible different behavior. The interference characteristics of a QS are based on the phases of different components. Thus the relative phase factors are critical but the overall phase factor is irrelevant.

Sep of variables

Schrodinger	Wave