Question

Demonstrate that the ground-state wave function for the one-dimensional harmonic oscillator satisfies the appropriate Schrodinger's equation

Answer 1

According to quantum mechanics, the time independent-one-dimensional Schrondinger's equation for simple simple harmonic oscillator is,

   .....( 1 )

The potential energy of the simple harmonic oscillator,

putting this in equation (1),

     .....( 2 )

or

Assume    and   

Then,      .....( 3 )

This is the Schrodinger equation for one dimensional simple harmonic oscillator.

Here is the coefficient of . So, it is difficult to obtain its solution. Hence, we will find its asymptotic solution.

When , this implies   . So, we can write

   .......( 4 )

We know solution of above equation,

Now we take ' - ' sign because it obeys the condition that decreases with increasing .

General Solution: ,

Differentiating with respect to x,

Again differentiating with respect to x,

Substituting values of and in equation ( 3 ),

  

   ........( 5 )

Now substituting and , converting into standard Hermite polynomial equation,

If , then

So,   

Substituting values of and in equation ( 5 ),

Now putting

   .......( 6 )

This is the standard hermite equation. It can be expressed as

     .........( 7 )

  

  

putting in equation ( 6 )

  

This expression is valid only if coefficient of each power of y is zero.

Therefore,   

   ....... ( 8 )

Then, complete solution of Schrondinger equation will be

......... ( 9 )

Solving equation ( 8 ) and equation ( 9 ) will provide us ground-state wave function,

   ,