130_notes.dvi

We cancompute the coefficientusing our operators.

|C|^2 = 〈A†un|A†un〉=〈AA†un|un〉

= 〈(A†A+ [A,A†])un|un〉= (n+ 1)〈un|un〉=n+ 1

The effect of theraising operatoris

A†un=

√

n+ 1un+1.

Similarly, the effect of thelowering operatoris

Aun=

√

nun− 1.

These are extremely important equations for any computation in the HO problem.

We can alsowrite any energy eigenstate in terms of the ground state and the raising
operator.

un=

1

√

n!

(A†)nu 0

10.4 Expectation Values ofpandx.

It is important to realize that we can just use the definition ofAto writexandpin terms of the
raising and lowering operators.

x =

√

̄h

2 mω

(A+A†)

p = −i

√

m ̄hω

2

(A−A†)

This will allow for any computation.

• 
  

  See Example 10.6.1:The expectation value ofxfor any energy eigenstate is zero.*

  
  


• 
  

  See Example 10.6.2:The expectation value ofpfor any energy eigenstate is zero.*

  
  


• 
  

  See Example 10.6.3:The expectation value ofxin the state√^12 (u 0 +u 1 ).*

  
  


• 
  

  See Example 10.6.4:The expectation value of^12 mω^2 x^2 for any energy eigenstate is^12

  
  

(

n+^12

)

̄hω.*

• See Example 10.6.5:The expectation value of p

2

2 mfor any energy eigenstate is

1

2

(

n+^12

)

̄hω.*

• See Example 10.6.6:The expectation value ofpas a function of time for the stateψ(t= 0) =
  √^1
  2 (u^1 +u^2 ) is−

√

m ̄hωsin(ωt).*

10.5 The Wavefunction for the HO Ground State

The equation
Au 0 = 0