Transcript PowerPoint 프레젠테이션

1. Bloch theorem
The 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 theorem
we 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
2. Time-dependent Schrödinger equation
wave equation with this relation
and with a solution of the form
between energy and frequency,
in a uniform potential
Schrödinger postulated the time-dependent 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
Solutions whose spatial behavior is steady in time should satisfy the time-independent equation
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
Quantum Mechanics (14/2)
CH. Jeong
2. Time-dependent Schrödinger equation
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
Quantum Mechanics (14/2)
CH. Jeong
2. Time-dependent Schrödinger equation
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)
Quantum Mechanics (14/2)
CH. Jeong
2. Time-dependent Schrödinger equation
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,
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 without
making any difference to the resulting oscillation.
Quantum Mechanics (14/2)
CH. Jeong
2. Time-dependent Schrödinger equation
Quantum Mechanics (14/2)
CH. Jeong
3. Propagating wave packets
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”
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
Quantum Mechanics (14/2)
CH. Jeong