Let us analyze the given problem. You are asked to prove that if the wave function $\Psi(\vec{r}, t)$ is normalized, then $\psi(\vec{r})$ is also normalized.

Definitions and Approach:

1. A wave function $\Psi(\vec{r}, t)$ is considered normalized if: $\int |\Psi(\vec{r}, t)|^2 \, d^3r = 1.$ This is the condition for normalization over all space.

2. The time-independent part of the wave function, $\psi(\vec{r})$, is extracted from the separation of variables: $\Psi(\vec{r}, t) = \psi(\vec{r}) \cdot e^{-iE t / \hbar},$ where $e^{-iE t / \hbar}$ is a purely time-dependent phase factor.

3. The goal is to prove that if $\Psi(\vec{r}, t)$ is normalized, then $\psi(\vec{r})$ satisfies its own normalization condition: $\int |\psi(\vec{r})|^2 \, d^3r = 1.$

Proof:

1. Start with the normalization of $\Psi(\vec{r}, t)$: Since $\Psi(\vec{r}, t)$ is normalized, we know: $\int |\Psi(\vec{r}, t)|^2 \, d^3r = 1.$

2. Substitute $\Psi(\vec{r}, t) = \psi(\vec{r}) \cdot e^{-iE t / \hbar}$: The squared magnitude of $\Psi(\vec{r}, t)$ is: $|\Psi(\vec{r}, t)|^2 = |\psi(\vec{r}) \cdot e^{-iE t / \hbar}|^2.$

3. Evaluate the modulus: The modulus of the phase factor $e^{-iE t / \hbar}$ is 1 (since it is a complex exponential). Thus: $|\Psi(\vec{r}, t)|^2 = |\psi(\vec{r})|^2 \cdot |e^{-iE t / \hbar}|^2 = |\psi(\vec{r})|^2.$

4. Substitute this result into the normalization condition: $\int |\Psi(\vec{r}, t)|^2 \, d^3r = \int |\psi(\vec{r})|^2 \, d^3r = 1.$

5. Conclusion: The normalization of $\Psi(\vec{r}, t)$ directly implies the normalization of $\psi(\vec{r})$, as the time-dependent factor does not affect the integral.

Final Answer:

$\int |\psi(\vec{r})|^2 \, d^3r = 1,$ proving that $\psi(\vec{r})$ is normalized if $\Psi(\vec{r}, t)$ is normalized.

Would you like additional clarifications or an example?
Here are 5 related questions to explore further:

1. What is the significance of the phase factor $e^{-iEt/\hbar}$ in quantum mechanics?
2. How does normalization relate to the probability interpretation of quantum mechanics?
3. Can a time-dependent wave function ever lose normalization? Under what circumstances?
4. What happens to the normalization condition if the system is not isolated?
5. How would you verify normalization for a 1D wave function $\psi(x)$?

Tip: Always verify whether the wave function satisfies boundary conditions when solving normalization problems.