130_notes.dvi

and

〈ψ|φ〉≡

∫∞

−∞

ψ∗(x)φ(x)dx

We use this shorthandDirac Bra-Ket notationa great deal.

1.7 Commutators

Operators (or variables in quantum mechanics) do not necessarily commute. We can compute the
commutator (See section 6.5) of two variables, for example

[p,x]≡px−xp=

̄h

i

Later we will learn to derive the uncertainty relation for two variables from their commutator. We
will also use commutators to solve several important problems.

1.8 The Schr ̈odinger Equation

Wave functions must satisfy the Schr ̈odinger Equation (See section 7) which is actually a wave
equation.
− ̄h^2
2 m

∇^2 ψ(~x,t) +V(~x)ψ(~x,t) =i ̄h

∂ψ(~x,t)

∂t

We will use it to solve many problems in this course. In terms of operators, this can be written as

Hψ(~x,t) =Eψ(~x,t)

where (dropping the (op) label)H= p

2
2 m+V(~x) is the Hamiltonian operator. So the Schr ̈odinger
Equation is, in some sense, simply the statement (in operators) that the kinetic energy plus the
potential energy equals the total energy.

1.9 Eigenfunctions, Eigenvalues and Vector Spaces

For any given physical problem, the Schr ̈odinger equation solutions which separate (See section 7.4)
(between time and space),ψ(x,t) =u(x)T(t),are an extremely important set. If we assume the
equation separates, we get the two equations (in one dimension forsimplicity)

i ̄h

∂T(t)

∂t

=E T(t)

Hu(x) =E u(x)

The second equation is called the time independent Schr ̈odinger equation. For bound states, there
are only solutions to that equation for some quantized set of energies

Hui(x) =Eiui(x).

For states which are not bound, a continuous range of energies is allowed.