130_notes.dvi

It shows that due to the wave nature of particles, we cannot localize a particle into a small volume
without increasing its energy. For example, we can estimate the ground state energy (and the size
of) a Hydrogen atom very well from the uncertainty principle.

The next step in building up Quantum Mechanics is to determine how a wave function develops with
time – particularly useful if a potential is applied. The differential equation which wave functions
must satisfy is called the Schr ̈odinger Equation.

1.5 Operators

The Schr ̈odinger equation comes directly out of our understanding of wave packets. To get from
wave packets to a differential equation, we use the new concept of(linear) operators (See section 6).
We determine the momentum and energy operators by requiring that, when an operator for some
variablevacts on our simple wavefunction, we getvtimes the same wave function.

p(xop)=

̄h

i

∂

∂x

p(xop)ei(~p·~x−Et)/ ̄h=

̄h

i

∂

∂x

ei(~p·~x−Et)/h ̄=pxei(~p·~x−Et)/ ̄h

E(op)=i ̄h

∂

∂t

E(op)ei(~p·~x−Et)/ ̄h=i ̄h

∂

∂t

ei(~p·~x−Et)/ ̄h=Eei(~p·~x−Et)/ ̄h

1.6 Expectation Values

We can use operators to help us compute the expectation value (See section 6.3) of a physical
variable. If a particle is in the stateψ(x), the normal way to compute the expectation value off(x)
is

〈f(x)〉=

∫∞

−∞

P(x)f(x)dx=

∫∞

−∞

ψ∗(x)ψ(x)f(x)dx.

If the variable we wish to compute the expectation value of (likep) is not a simple function ofx, let
its operator act onψ(x)

〈p〉=

∫∞

−∞

ψ∗(x)p(op)ψ(x)dx.

We have a shorthand notation for the expectation value of a variablevin the stateψwhich is quite
useful.

〈ψ|v|ψ〉≡

∫∞

−∞

ψ∗(x)v(op)ψ(x)dx.

We extend the notation from just expectation values to

〈ψ|v|φ〉≡

∫∞

−∞

ψ∗(x)v(op)φ(x)dx