Lecture 3: Review of quantum mechanics
¯h/p
{Ψ}
¯h/p
{Ψ}Ψ∗Ψ
x
{Ψ}{Ψ}Ψ∗Ψ
x
We postulate that
• the state of the system is completely described by a wave function.
• if possible states of our system are described by ψ1, ψ2, . . ., their linear
combination also describes a possible state of the system.
• any measurable property is represented by an operator (acting on
the wave function) and that result of a measurement must be one of
eigenvalues of the operator.
• the expected result of measuring a quantity A represented by an
operator ˆA in a state of the system described by a wave function Ψ is
A = Ψ| ˆA|Ψ .
• that if Am is measured in the state described by |Ψ , then the state
immediately after the measurement is described by ˆPm|Ψ / Ψ| ˆPm|Ψ ,
where ˆPm is the projection operator associated with Am.
• that operators of position and momentum obey the relations
[ˆrj, ˆpk] = i¯hδjk; [ˆrj, ˆrk] = [ˆpj, ˆpk] = 0.
• that evolution of a system in time is given by the Hamiltonian:
i¯h∂Ψ
∂t = ˆHΨ.
i¯h
∂Ψ
∂t
=


−
¯h2
2m



∂
∂x
+ QAx
2
+


∂
∂y
+ QAy


2
+
∂
∂z
+ QAz
2


 + QV (x, y, z)



ˆH
Ψ
[ˆLx, ˆLy] = i¯hˆLz
[ˆLy, ˆLz] = i¯hˆLx
[ˆLz, ˆLx] = i¯hˆLy
[ˆL2, ˆLx] = 0
[ˆL2, ˆLy] = 0
[ˆL2, ˆLz] = 0