Harmonic Oscillator Eigenfunctions

 Simplifying the Differential Equation for the Harmonic Oscillator

 The annihilation and creation operators are defined, respectively, as

 where Let  where i.e. K = mw₀², m is the mass of the particle and K is the spring constant. In terms of = x and  Eq. (1) becomes

 To simplify Eq. (2) let

 In terms x Eq. (2) becomes

The time-independent Schrodinger equation for the harmonic oscillator, in terms of the creation and annihilation operators is given by

 To evaluate the quantity in brackets first express in terms defined in Eq. (4), i.e.

 Substituting Eq. (6) into Eq. (5) gives the simplified differential equation for

 Finding the Eigenfunction corresponding to the Ground State

 Returning to the annihilation operator  the equation , in terms of Eq. (4), becomes

 We can choose the sign of to be positive since this amounts to multiplication by an arbitrary phase factor. Therefore

A₀ is chosen such that , i.e.

 Note:

 Therefore  becomes

can also be expressed in terms of x as follows.

 As above with A₀ the constant B₀ is chosen such that  when  is expressed in terms of the variable x. Therefore

 Recall that x = bx � that dx = bdx  �  dx = dx/b. Eq. (13)   then becomes

Solving for B₀ gives

 Finding c_n

 Notation:  Let  be used in what follows.

 As shown elsewhere is proportional to . I.e.

 Our next task is to determine c1 which is determined as usual  by demanding . Note that . Thus

The arbitrary phase factor will always be chosen to have the value of unity.  Similarly is proportional to . I.e.

Therefore

 In general

We therefore have

Eq. (22) then becomes

Once again we follow the same procedure above for Eq. (21) but now for n = 3 to get

 This process can be continued. The general results are therefore given by

 In terms of the eigenfunctions we have

where

 When the operator acts on exp(-x²/2) results in an nth order polynomial multiplied by exp(-x²/2), i.e.

 Hn(x) are the Hermite polynomials.

 The first 5 eigenfunctions are listed below 

 A Few Useful Relations

 Two useful relations are

 Eq. (31a) is readily derived as follows. We have from Eq. (27)

 Eq. (31b) obtained by multiplying Eq. (25) by to give