to agree with the opposite sign conventions for spatial and time translations in
the definitions of momentum and energy.
One finds for free particle solutions
ψ̂(k,ω) =√^1
2 π
∫∞
−∞
eiωte−i
21 mk^2 t ̃
ψ(k,0)dt
=δ(ω−
1
2 m
k^2 )
√
2 πψ ̃(k,0)
soψ̂(k,ω) will be a distribution onk−ωspace that is non-zero only on the
parabolaω= 21 mk^2. The space of solutions can be identified with the space of
functions (or distributions) supported on this parabola. Energy eigenstates of
energyEwill be distributions with a dependence onωof the form
ψ̂E(k,ω) =δ(ω−E)ψ ̃E(k)
For free particle solutions one hasE= k
2
2 m.
ψ ̃E(k) will be a distribution ink
with a factorδ(E−k
2
2 m).
For any functionf(k), the delta function distributionδ(f(k)) depends only
on the behavior offnear its zeros. Iff′ 6 = 0 at such zeros, one has (using linear
approximation near zeros off)
δ(f(k)) =
∑
kj:f(kj)=0
δ(f′(kj)(k−kj)) =
∑
kj:f(kj)=0
1
|f′(kj)|
δ(k−kj) (11.9)
Applying this to the case off(k) =E−k
2
2 m, with a graph that has two zeros,
atk=±
√
2 mEand looks like
√
− 2 mE
√
2 mE
E
k
f(k) =E−
k^2
2 m
Figure 11.1: Linear approximations near zeros off(k) =E−k
2
2 m