Physics · Advanced university
Expectation value in an orthogonal eigenbasis
The squared amplitudes are probabilities. Orthogonality and normalization are specified.
Method and conditions
The squared amplitudes are probabilities. Orthogonality and normalization are specified.
⟨A⟩ = Σ |cj|² aj
Worked examples
A normalized state has probabilities 0.3 and 0.7 for two orthogonal eigenstates of A with eigenvalues -4 and 18. Calculate ⟨A⟩. Round the final result to 4 decimal places if needed.
The expectation is the probability-weighted sum 0.3(-4) + (1−0.3)(18).
11.4
A normalized state has probabilities 0.1 and 0.9 for two orthogonal eigenstates of A with eigenvalues -1 and 16. Calculate ⟨A⟩. Round the final result to 4 decimal places if needed.
The expectation is the probability-weighted sum 0.1(-1) + (1−0.1)(16).
14.3
Try it
A normalized state has probabilities 0.8 and 0.19999999999999996 for two orthogonal eigenstates of A with eigenvalues 10 and 27. Calculate ⟨A⟩. Round the final result to 4 decimal places if needed.
A normalized state has probabilities 0.3 and 0.7 for two orthogonal eigenstates of A with eigenvalues 8 and 20. Calculate ⟨A⟩. Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 13.4
The expectation is the probability-weighted sum 0.8(10) + (1−0.8)(27).
- 16.4
The expectation is the probability-weighted sum 0.3(8) + (1−0.3)(20).
Further reading and review references
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