Chapter 3: 3.15P (page 113)

Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if $\hat{P}$ and $\hat{Q}$ have a complete set of common eigenfunctions, then $[\hat{P} \cdot \hat{Q}] f = 0$ for any function in Hilbert space.

Short Answer

Expert verified

A whole set of common eigenfunctions cannot be shared by two noncommuting operators.

Step by step solution

Concept used

The same complete set of common eigenfunctions:

$\hat{P} f_{n} = λ_{n} f_{n} and \hat{Q} f_{n} = μ_{n} f_{n}$

Calculation

Assuming the operators $\hat{P}$ and $\hat{Q}$ have the same complete set of common eigenfunctions, that is:

$\hat{P} f_{n} = λ_{n} f_{n} and \hat{Q} f_{n} = μ_{n} f_{n}$

And suppose the set $f_{n}$ is complete, so that any function in Hilbert space $f (x)$ can be expressed as a linear combination, that is:

$f = \sum c_{n} f_{n}$

Solving for the above function,

$[\hat{P}, \hat{Q}] f = (\hat{P} \hat{Q} - \hat{Q} \hat{P}) \sum c_{n} f_{n} = \hat{P} (\sum c_{n} μ_{n} f_{n}) - \hat{Q} (\sum c_{n} λ_{n} f_{n}) = \sum c_{n} μ_{n} λ_{n} f_{n} - \sum c_{n} λ_{n} μ_{n} f_{n} = 0$

If two operators have the same set of eigenfunctions, the commutator will be zero.

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Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if $\hat{P}$ and $\hat{Q}$ have a complete set of common eigenfunctions, then $[\hat{P} \cdot \hat{Q}] f = 0$ for any function in Hilbert space.

Short Answer

Step by step solution

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Calculation

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Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if P^and Q^have a complete set of common eigenfunctions, then[P^·Q^]f=0 for any function in Hilbert space.

Short Answer

Step by step solution

Concept used

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Measurements

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Study anywhere. Anytime. Across all devices.

Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if $\hat{P}$ and $\hat{Q}$ have a complete set of common eigenfunctions, then $[\hat{P} \cdot \hat{Q}] f = 0$ for any function in Hilbert space.