Question

Evaluate the expectation value of r for the wave function (πa^3)^(-1/2)exp (-r/a)?

Answer 1

Answer on Question #81260 Physics / Optics

Evaluate the expectation value of $r$ for the wave function $\psi(\mathbf{r}) = (\pi a^3)^{-\frac{1}{2}} \exp(-r/a)$.

Solution:

The expectation value

$$\langle r \rangle = \int \psi^*(\mathbf{r}) r \psi(\mathbf{r}) d\mathbf{r} = \frac{1}{\pi a^3} \int \exp\left(-\frac{2r}{a}\right) r d\mathbf{r} = \frac{4\pi}{\pi a^3} \underbrace{\int_0^\infty \exp\left(-\frac{2r}{a}\right) r^3 dr}_{3a^4/8} = \frac{4}{a^3} \times \frac{3a^4}{8} = \frac{3}{2} a$$

Answer: $\langle r \rangle = \frac{3}{2} a$

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