130_notes.dvi

6 Operators

Operators will be used to help us derive a differential equation that our wave-functions must satisfy.
They will also be used in almost any Quantum Physics calculation.

An example of a linear operator (See section 7.5.1) is a simple differential operator like∂x∂, which
we understand to differentiate everything to the right of it with respect tox.

6.1 Operators in Position Space

To find operators for physical variables in position space, we will lookat wave functions with definite
momentum. Our state of definite momentump 0 (and definite energyE 0 ) is

up 0 (x,t) =

1

√

2 π ̄h

ei(p^0 x−E^0 t)/ ̄h.

We can build any other state from superpositions of these states using the Fourier Transform.

ψ(x,t) =

∫∞

−∞

φ(p)up(x,t)dp

6.1.1 The Momentum Operator

We determine the momentum operator by requiring that, when we operate withp
(op)
x onup 0 (x,t),
we getp 0 times the same wave function.

p(op)up 0 (x,t) =p 0 up 0 (x,t)

This means that for these definite momentum states, multiplying byp(xop)is the same as multiplying
by the variablep. We find that this is true for the followingmomentum operator.

p(op)=

̄h

i

∂

∂x

We can verify (See section 6.6.1) that this works by explicit calculation.

If we take our momentum operator andact on a arbitrary state,

p(op)ψ(x,t) =p(op)

∫∞

−∞

φ(p)up(x,t)dp=

∫∞

−∞

φ(p)p up(x,t)dp

it gives us the rightpfor each term in the integral. This will allow us to compute expectationvalues
for any variable we can represent by an operator.