130_notes.dvi

The expectation values of physical quantities should be real.

Gasiorowicz Chapter 3

Griffiths Chapter 1

Cohen-Tannoudji et al. Chapter

6.4 Dirac Bra-ket Notation

A state with definite momentump.|p〉
A state with definite positionx.|x〉
The“dot product”between two abstract statesψ 1 andψ 2.

〈ψ 1 |ψ 2 〉=

∫∞

−∞

ψ 1 ∗ψ 2 dx

This dot product projects the stateψ 2 ontoψ 1 and represents the amplitude to go fromψ 2 toψ 1.

To find the probability amplitude for our particle to by at any positionx, we dot the state of definite
xinto our stateψ.ψ(x) =〈x|ψ〉

To find the probability amplitude for our particle to have a momentump, we dot the state of definite
xinto our stateψ.φ(p) =〈p|ψ〉

6.5 Commutators

Operators (or variables in quantum mechanics) do not necessarily commute. We can see our first
example of that now that we have a few operators. Wedefine the commutatorto be

[p,x]≡px−xp

(usingpandxas examples.)

We will nowcompute the commutatorbetweenpandx. Becausepis represented by a differential
operator, we must do this carefully. Lets think of the commutatoras a (differential) operator too, as
generally it will be. To make sure that we keep all the∂x∂ that we need, we will compute [p,x]ψ(x)
then remove theψ(x) at the end to see only the commutator.

[p,x]ψ(x) = pxψ(x)−xpψ(x) =

̄h

i

∂

∂x

xψ(x)−x

̄h

i

∂

∂x

ψ(x)