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Chapter 40: Q. 38 (page 1176)

Show that the constant b used in the quantum-harmonic-oscillator wave functions (a) has units of length and (b) is the classical turning point of an oscillator in the n=1ground state.

Short Answer

Expert verified

All the statements are proved.

Step by step solution

01

Part (a) Step 1: Given information 

We have given,

System of quantum harmonic oscillator.

We have to find the unit of constant b.

02

Simplify

Since we know that the value of b is given by,

b=h2蟺尘蝇

Where

h has unit J.s and m has unit of kg and w is in per second then,

bunit=ML2T-1MT-112bunit=L

Hence proved.

03

Part (b) Step 1: Given information

We have to show that b is classical turning point for ground state.

04

Simplify

To show that the classical turning point is b is ,

E=12kx2=12m蝇2A2A=2Em蝇2A=2m蝇2(n+1)h蝇2蟺A=(2n+1)h2蟺m蝇for,n=0A=h2蟺m蝇=bb=x

Hence proved.

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