Quantum Mechanics (14/2) CH. Jeong

1. Bloch theoremThe wavefunction in a (one-dimensional) crystal can be written in the form

subject to the condition

Multiply by to obtain

Hence if we define a function

we can restate Eq. as

and hence u(x) is periodic with the lattice periodicity.

Quantum Mechanics (14/2) CH. Jeong

1. Bloch theoremwe can rewrite the Bloch theorem equation in the alternative form

where u(x) is periodic with the lattice periodicity.

Concept of the Bloch functions. We can think of the exp(ikx) as being an example of an “envelope” function that multiplies the unit cell function u(x)

Quantum Mechanics (14/2) CH. Jeong

wave equation with this relation between energy and frequency,

and with a solution of the form in a uniform potential

Solutions whose spatial behavior is steady in time should satisfy the time-independent equation

Suppose that we had a solution where the spatial behavior of the wavefunction did not change its form with time. We could allow for a time-varying multiplying factor, A(t) , in front of the spatial part of the wavefunction, i.e., we could write

Adding the factor A(t) in front of ψ(r) makes no difference in Eq. above Ψ(r,t) would also be a solution, regardless of the form of A(t) , i.e., we would have

2. Time-dependent Schrödinger equation

Schrödinger postulated the time-dependent equation

Quantum Mechanics (14/2) CH. Jeong

Substituting the form ψ(r,t) =A(t)ψ(r) into the time-dependent Schrödinger equation

(presuming the potential V is constant in time) then gives

So

i.e., for some constant Ao

Hence, if the spatial part of the wavefunction is steady in time the full time-dependent wavefunction can be written in the form

2. Time-dependent Schrödinger equation

Quantum Mechanics (14/2) CH. Jeong

Suppose we expand the original spatial solution at time t=0 in energy eigenfunctions,

where the an are the expansion coefficients

(the an are fixed complex numbers). Any spatial function ψ(r) can be expanded this way because of the completeness of the eigenfunctions ψn(r)

We can now write a corresponding time-dependent function

We know this is a solution to the time-dependent Schrödinger equation because it is made up from a linear combination of solutions to the equation.

As a check, at t = 0 this correctly gives the known spatial form of the solution.This is the solution to the time-dependent Schrödinger equation (for the case where V does not vary in time)

2. Time-dependent Schrödinger equation

Quantum Mechanics (14/2) CH. Jeong

Suppose we have an infinite potential well (i.e., one with infinitely high barriers), and that the particle in that well is in a (normalized) linear superposition state with equal parts of the first and second states of the well,

2. Time-dependent Schrödinger equation

Then the probability density is given by

This probability density has a part that oscillates at an angular frequency

Note that the absolute energy origin does not matter here.We could have added an arbitrary amount onto both of the two energies E1 and E2 with-out making any difference to the resulting oscillation.

Quantum Mechanics (14/2) CH. Jeong

2. Time-dependent Schrödinger equation

Quantum Mechanics (14/2) CH. Jeong

To understand movement, we have to construct a “wave-packet” – a linear superposition of waves that adds up to give a “packet” that is approximately localized in space at any given time. To understand what behavior we expect from such packets, we have to introduce the concept of group velocity.

Elementary wave theory says the velocity of the center of a wave packet or pulse is the “group velocity”

where ω is the frequency and k is the wave vector.

which can be viewed as an underlying wave

This envelope can be seen to move at a the “group velocity”

3. Propagating wave packets

Quantum Mechanics (14/2) CH. Jeong

3. Propagating wave packets For a particle such as an electron, phase velocity and group velocity of quantum mechanical waves are almost never the same. For the simple free electron, the frequency ω is not proportional to the wave vector magnitude k. the time-independent Schrödinger equation tells us that, for any wave component In fact (for zero potential energy),

i.e., So

We see then that the propagation of the electron wave is always highly dispersive. Hence, we have a velocity for a wavepacket made up out of a linear superposition of waves of energies near E,

so that