Page 106 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Quantum Mechanics of Single Electrons
ˆ
(
2m)∇
— ⁄
2
2
ond order with its coefficient T =
is, according to the dis-
–
cussion about the momentum operator in 3.1.1, the operator of the kinetic
energy in a spatial representation. The equation of motion for the
Schrödinger field Ψ in a potential is given by
∂
ˆ
ˆ
(
,
(
,
(
,
i— Ψ x t) = ( T + U x())Ψ x t) = HΨ x t) (3.15)
t ∂
ˆ
ˆ
H = T + U x() is called the Hamilton operator of a closed system if
U x() does not depend explicitly on the time variable . This means that
t
the solution of the system described by (3.15) is invariant under transla-
tions in time. Wavefunctions for which the energy has a specific value are
called stationary states. We denote them as Ψ and call them eigenfunc-
n
tions of the Hamilton operator. All of them follow the equation
ˆ
HΨ = E Ψ , which is called the eigenvalue equation for the Hamilton
n n n
operator. The total of eigenfunctions and eigenvalues is called the spec-
trum of the Hamilton operator. In this case we integrate (3.15) in time,
which gives
 i 
Ψ = exp  – ---E t ψ x() (3.16)

n
n
n
—
where the function ψ depends only on the coordinates. (3.16) gives the
n
time dependence of the wavefunction, while the spatial dependence and
the energy of the eigenstate must be found solving the respective eigen-
value problem resulting inserting (3.16) into (3.15)
ˆ
Hψ x() = E ψ x() (3.17)
n n n
Quantum The numbers n counting the different energies are called quantum num-
Numbers bers. The stationary state with the lowest energy we call the ground state.
We recall some technical aspects from linear algebra:
• For every discrete eigenvalue problem, there might be multiple eigen-
values which results in different eigenvectors for the same eigenvalue.

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