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 $n = 1$ ground state.

Short Answer

Expert verified

All the statements are proved.

Step by step solution

Part (a) Step 1: Given information

We have given,

System of quantum harmonic oscillator.

We have to find the unit of constant b.

Simplify

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

$b = \sqrt{\frac{h}{2 πmω}}$

Where

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

$b unit = {(\frac{{ML}^{2} T^{- 1}}{{MT}^{- 1}})}^{\frac{1}{2}} b unit = L$

Hence proved.

Part (b) Step 1: Given information

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

Simplify

To show that the classical turning point is b is ,

$E = \frac{1}{2} k x^{2} = \frac{1}{2} m ω^{2} A^{2} A = \sqrt{\frac{2 E}{m ω^{2}}} A = \sqrt{\frac{2}{m ω^{2}} (n + 1) \frac{h ω}{2 π}} A = \sqrt{(2 n + 1) \frac{h}{2 π m ω}} f o r, n = 0 A = \sqrt{\frac{h}{2 π m ω}} = b b = x$

Hence proved.

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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 $n = 1$ ground state.

Short Answer

Step by step solution

Part (a) Step 1: Given information

Simplify

Part (b) Step 1: Given information

Simplify

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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

Step by step solution

Part (a) Step 1: Given information

Simplify

Part (b) Step 1: Given information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Further Mechanics and Thermal Physics

Fields in Physics

Translational Dynamics

Particle Model of Matter

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

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 $n = 1$ ground state.