(a)
The energy of three-dimensional harmonic oscillator using separation of variable.
(a)
Answer to Problem 4.46P
Three-dimensional harmonic oscillator can be rewritten as three one-dimensional oscillators and its energy is
Explanation of Solution
Write the expression for Schrodinger equation for harmonic oscillator in terms of polar co-ordinates
Here,
Write the expression for the change of variable
Rewrite expression (II) using (I)
The first term is a function of
Rewrite expression (III) as three separate differential equations
Rewrite the expression (III)
Conclusion:
Expressions (IV), (V) and (VI) are of the form of one-dimensional oscillator
Write the expression for energy of one-dimensional oscillators
Here,
Rewrite expression (VII)
Let
Therefore, the energy of the three-dimensional harmonic oscillator is
(b)
The degeneracy of the energy.
(b)
Answer to Problem 4.46P
The degeneracy of energy is
Explanation of Solution
Consider the case when
Consider the case when
Consider the case when
Conclusion:
Generalise the number of ways in which the energy can be distributed among the three levels
Here,
Therefore, the degeneracy of energy is
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Chapter 4 Solutions
Introduction To Quantum Mechanics
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