STATES AS VECTORS:

• Each physical state corresponds to a state vector |a> ("ket"). Different physical states correspond to different vectors and vice versa. "a" is a label.
  
  
• State vectors obey the properties of vectors, see list p98 for addition and mult by const. Also
  
  • If |a> and |b> are possible state vectors, then A|a> +B|b> is also a possible state vector.
    
    
    
• A vector space must have a norm or "length."
  • "Inner product" of state vector with itself, written <a|a>, gives the norm.
    
    
    
  • <a|a> = 1. Normalization condition.
    
    
  
  
  
• Mathematical properties of inner product
  
  
  • it is a (complex) number
    
    
  • see list p99, including <a|b> = <b|a>*
  
  
  
  
  
• Physical meaning of inner product <b|a>:
  |<b|a>|2 = |<a|b>|² = probability of finding particle in state |a> to be in state |b> (or vice versa).
  
  
  
  • psi(x) = <x|a>
    |psi(x)|² = |<x|a>|² = probability of finding particle in state |a> to be in state |x> --
    in other words the probability density of finding particle in state |a> to be at position x.
    
    
    
  • phi(p) = <p|a>
    |phi(p)|² = |<p|a>|² = probability of finding particle in state |a> to be in state |p> --
    in other words the probability density of finding particle in state |a> to have momentum p.
    
    
    
  • a_n = <E_n|a>
    
    |a_n|² = |<E_n|a>|² = probability of finding particle in state |a> to be in state |E_n> --
    in other words the probability of finding particle in state |a> to have energy E_n.
  
  
  
  
  
  
  
  
• The set of all eigenstates of an observable form a complete orthonormal set of states, {|ai>}.*
  • ortho- means orthogonal:
    If |a_n> and |a_m> are eigenstates corresponding to different eigenvalues, then <a_n|a_m>=0.
    • Ex/ Suppose |E₁> is the ground state and |E₂> is first excited state.
      
      The probability that a particle in the ground state would be found in the first excited state is zero.
      |<E₁|E₂>|² = 0, so <E₁|E₂> = 0.
    
    
    
  • -normal means normalized: <a_i|a_i> = 1 for all i.
  • complete means it can be used as a basis for a representation: Any state vector can be written as a linear combo of these basis vectors:
    • Ex/ Suppose |a> cannot be written as linear combo of energy eigenstates. Suppose it is
      |a> = (3)^-1/2 |E1> + (3) ^-1/2 |E2> + (3) ^-1/2 |not an energy eigenstate>,
      
      
      then with prob 1/3 an energy measurement will result in "not an energy"!! Can't be.
    
    
    
    
    
  
  
  
• There are many observables whose eigenstates we can use as basis vectors, so there are many representations of a state vector:
  
  
  • Position representation:
    
    
    
    
    
    
  • where |x> is an eigenstate of position - state of definite position, x.
  • Momentum representation:
    
    
    
    
    
    
  • where |p> is an eigenstate of momentum - state of definite momentum, p.
  • Energy representation:
    
    
    
    
    
    
  • where |E_n> is an eigenstate of energy - state of definite energy E_n.

OBSERVABLES AS OPERATORS

Recall for all operators: Oop|a> = |a'>

and for linear operators: Oop ( |a> + |b> ) = Oop|a> + Oop|b>

unit operator: 1op|a> =|a> for all states

Matrices have these properties when applied to vector (column matrix). Some operators are represented by square matrices in a particular basis.

Note: <b|Oop|a> means <b| ( Oop |a> )

<b|Oop|a> = ( <b| Oop*T ) |a>

Ex//

|a> =

0   

1   

|b> =

i   

0   

Oop =

1     0                                                             

i     1                                                             

If Oop corresponds to an observable, then Oop*T = Oop . Such an operator is Hermitian.

EIGENSTATES AND EIGENVALUES

Oop|n> = On |n> where On is a number

If operator is represented by matrix, you get a matrix eigenvalue problem.

Ex//

Eigenvalues of operator represented by

5      2     1       

3      3     3       

1      6     2       

Suppose we found all eigenstates |i> of an obsevable, with eigenvalues Oi.

Suppose the state of our particle is |a>.

The probability that the the observable is found to have value Oi is |<i|a>|².

The probability that the observable is found to have any value is

true for any state vector |a> and any complete set {|i>}.

We can write any state vector as a combo of these states:

Check that <a|a> =1.

COMMUTATOR

[Oop,Qop] =OopQop - QopOop

or OopQop = QopOop +[Oop,Qop]

Ex// Find <x1|popxop|p1> (a number depending on the specific position x1 and the specific momentum p1).

Ex// Find commutator of Oop above and

Qop =

2     1                                                               

1     1