1.12: Expectation values of position

Given that P(x) is the probability density of the electron at position x, we can determine the average, or expectation value of x from

\[ \langle x\rangle =\frac{\int^{+\infty}_{-\infty} xP(x)dx}{\int^{+\infty}_{-\infty} P(x)dx} \nonumber \]

Of course if the wavefunction is normalized then the denominator is 1.

We could also write this in terms of the wavefunction

\[ \langle x\rangle =\frac{\int^{+\infty}_{-\infty} x|\psi(x)|^{2} dx}{\int^{+\infty}_{-\infty} |\psi(x)|^{2}dx} \nonumber \]

Where once again if the wavefunction is normalized then the denominator is 1.

Since \(|\psi(x)|^{2} = \psi(x)^{*}\psi(x)\),

\[ \langle x\rangle =\frac{\int^{+\infty}_{-\infty} \psi(x)^{*}x\psi(x) dx}{\int^{+\infty}_{-\infty}\psi(x)^{*}\psi(x) dx} \nonumber \]

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