Colby College
Postulates of Quantum Mechanics
1
I.
The physical state of the system is described by a wave function as completely as possible.
The wave function is derived from an orthonormal set of eigenfunctions of the Hamiltonian.
Example
:
ParticleinaBox:
Ψ
n
(x) = (2/a)
½
sin(n
π
x/a)
are the solutions to
H
^
Ψ=ΕΨ
II.
Any observable may be represented by a linear operator. The results should be a real number.
The least
restrictive requirement is that the operator must be Hermitian:
∫
Ψ
*
j
o
^
Ψ
i
dx =
∫
Ψ
i
(o
^
Ψ
j
)
*
dx =
∫
Ψ
i
o
^*
Ψ
*
j
dx
The observable operator is constructed from the following table:
Classical
Quantum Operator
x
x
^
p
h
–
i
∂
∂
x
= – ih

∂
∂
x
t
t
^
E vs. time
E
^
= ih

∂
∂
t
III.
If the wave function is an eigenfunction of the observable, then repeated measurements of
the observable always give the same result, the eigenvalue:
if
o
^
Ψ
= o
Ψ
then each measurement gives the result, o.
Example
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 wave function, Ψn dx, quantum operator, px Ψn dx

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