Introduction to Quantum Mechanics · Problem 4.50

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · momentum_space_hydrogen

Problem

The (time-independent) momentum space wave function in three dimensions is defined by the natural generalization of Equation 3.54:

\phi(\mathbf{p}) \equiv \frac{1}{(2\pi\hbar)^{3/2}}\int e^{-i(\mathbf{p}\cdot\mathbf{r})/\hbar}\psi(\mathbf{r})\,d^3\mathbf{r}

. (4.223) (a) Find the momentum space wave function for the ground state of hydrogen (Equation 4.80). Hint: Use spherical coordinates, setting the polar axis along the direction of

\mathbf{p}

. Do the

\theta

integral first. Answer:

\phi(\mathbf{p}) = \frac{1}{\pi}\left(\frac{2a}{\hbar}\right)^{3/2}\frac{1}{\left[1 + (ap/\hbar)^2\right]^2}

. (4.224) (b) Check that

\phi(\mathbf{p})

is normalized. (c) Use

\phi(\mathbf{p})

to calculate

\langle p^2\rangle

, in the ground state of hydrogen. (d) What is the expectation value of the kinetic energy in this state? Express your answer as a multiple of

E_1

, and check that it is consistent with the virial theorem (Equation 4.218).

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