Concept explainers
(a)
Prove that for integers
(a)
Answer to Problem 5.38P
It has been proved that for integers
Explanation of Solution
Consider
Since,
The denominator vanishes only when
For
For
Conclusion:
It has been proved that for integers
(b)
The commutation relations for the ladder operators,
(b)
Answer to Problem 5.38P
The commutation relations for the ladder operators,
Explanation of Solution
Given the ladder operator
Solving for
Dropping the two terms involving commutators of two coordinates or two derivatives. The remaining commutators are
Hence it is proved that
Similarly solving for
Dropping the two terms involving commutators of two coordinates or two derivatives. The remaining commutators are
Similarly solving for
Dropping the two terms involving commutators of two coordinates or two derivatives. The remaining commutators are
Equating the two commutator,
Conclusion:
Hence it is proved that the commutation relations for the ladder operators,
(c)
Show that
(c)
Answer to Problem 5.38P
It is proved that
Explanation of Solution
Solving for
Since
Using the above equation to solve for
Further solving,
Since,
Hence the first relation is proved,
Solving for
Since
Using the above equation to solve for
Further solving,
Since,
Hence it is proved that
Conclusion:
It is proved that
(d)
Show that
(d)
Answer to Problem 5.38P
It has been proved that
Explanation of Solution
Solving for
Squaring on both sides,
Solving for
Substitute equation (IV) and (V) in
Solving for
The middle term in the above equation vanish when it is summed over
Adding the results
Conclusion:
It has been proved that
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Chapter 5 Solutions
Introduction To Quantum Mechanics
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