Harmonic Oscillator

 The Hamiltonian for the harmonic oscillator is given by setting

in the expression for the Hamiltonian giving

where K is the spring constant. is the operator for the x-component of momentum. The time-independent Schrodinger equation (for one-dimension)

 becomes 

Substituting  and = x into Eq. (2) gives

 The potential is shown in the diagram below

As seen in the above diagram the classical region is given when the kinetic energy is positive, i.e. for E > V. This corresponds to the interval –x₀ < x < x₀.

The Hamiltonian in Terms of the Creation and Annihilation Operators

 Let  where , i.e. K = mw₀². Define the operators

The relation will prove useful and is derived here.  

Substituting the relation results in  

 Note also that

 The inverses of Eq. (5) are found to be  

 The Hamiltonian in Eq. (1) can be expressed in terms of the creation and annihilation operators as follows.  Squaring both  and gives

Adding Eqs. (10) and (11) gives 

 This can be simplified by substituting in Eq. (8)

 Substituting  yields the final result 

At this point it will prove convenient to define the following operator

 With this definition the Hamiltonian in Eq. (14) becomes

 Let the eigenvalues of Hamiltonian be n and the eigenkets be . From Eq. (16) and Eq. (8) we get the following relation

  It follows from Eq. (17) that   is an eigenket of corresponding to the eigenvalue n - 1. That is to say

 Notice the effect that  has on an eigenket of ; It reduces the eigenvalue by one.  E.g.

 It is for this reason that  is called the annihilation operator.  In a similar fashion

 It follows from Eq. (20) that  is an eigenket of corresponding to the eigenvalue n + 1. That is to say

 Notice the effect that  has on an eigenket of ; It increases the eigenvalue by one.  E.g.

 It is for this reason that  is called the creation operator. 

Eigenvalues of the Hamiltonian

The expectation of the Hamiltonian, e.g. Eq. (1), in the state  is found to be

Note that for any Hermitian operator  the expectation of the square of  in the state  is given by

 And therefore the expectation of the square is always greater than or equal to zero. Thus Eq. (24) shows that . Therefore

Eq. (26) implies that all eigenstates corresponding to n < -1/2 must vanish.  This situation is obtained if we set

Eq. (27) implies that

 Applying to Eq. (27) and noting the definition  it follows that

Therefore the eigenvalue of  corresponding to  is zero. Multiplying the relation by gives

Or since

Therefore the eigenvalue of  corresponding to  is one. In a similar fashion it can be shown that the eigenvalue of  corresponding to  is n where n is an integer. The eigenvalues of the Hamiltonian are therefore found as follows

Therefore the possible values of the energy for the quantum harmonic oscillator are