Quantum Mechanics for Mathematicians

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