Expectation-Value Examples

Expectation values summarize repeated measurements on identically prepared systems. The calculation changes form with representation, but the physical question is always: which observable is being averaged in which state?

Core Forms

For a normalized pure state,

$$\langle A\rangle_\psi=\langle\psi\rvert A\lvert\psi\rangle.$$

In position representation for a multiplication operator $f(x)$,

$$\langle f(x)\rangle =\int dx\,\psi^*(x)f(x)\psi(x).$$

For a density matrix,

$$\langle A\rangle_\rho=\operatorname{Tr}(\rho A).$$

Use Expectation Value for the compact formula card and Trace Rule Expectation Values for the density-matrix derivation.

Example Routes

Need	Learn	Main Check
Wavefunction average	Expectation Values in Wave Mechanics	Operator representation and boundary behavior
Born-rule average	Expectation Values	Measurement outcomes and probabilities
Oscillator energy probabilities	Number States	Expansion coefficients in the energy basis
Two-state observable	Two-State Hamiltonians	Basis and normalization
Density-matrix trace	Density Matrix Expectation	Trace cyclicity and tensor factor
Variance	Variance	Compute $\langle A^2\rangle-\langle A\rangle^2$, not an average absolute deviation

Minimal Worked Check

For a normalized two-state vector

$$\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle, \qquad \lvert\alpha\rvert^2+\lvert\beta\rvert^2=1,$$

and observable $Z=\lvert0\rangle\langle0\rvert-\lvert1\rangle\langle1\rvert$,

$$\langle Z\rangle =\lvert\alpha\rvert^2-\lvert\beta\rvert^2.$$

This example is often safer than a wavefunction integral when the main issue is measurement basis rather than analysis.

Common Mistakes

• Computing $\langle A\rangle$ before checking that the state is normalized.
• Forgetting that momentum-space and position-space forms of the same operator look different.
• Using the diagonal entries of $\rho$ in the wrong basis.
• Confusing $\langle A\rangle$ with a possible measurement outcome.
• Computing $\langle A\rangle^2$ instead of $\langle A^2\rangle$ when finding a variance.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. In the two-state example above, what is $\langle Z\rangle$ for $\alpha=1/\sqrt{3}$ and $\beta=\sqrt{2/3}$?

Solution

The expectation value is $1/3-2/3=-1/3$. The possible outcomes are still $+1$ and $-1$; $-1/3$ is the ensemble average.