State the time-independent Schrodinger equation in three dimensions.

Write the three-dimensional Laplacian in Cartesian coordinates.

What is the probability density for a three-dimensional wave function?

State the normalization condition for a three-dimensional bound-state wave function.

What operator replaces the one-dimensional second derivative in the three-dimensional Schrodinger equation?

For a stationary state \(\psi(\vec r,t)=\phi(\vec r)e^{-iEt/\hbar}\), why is the probability density time independent?

For a free particle with \(\phi=e^{i\vec k\cdot\vec r}\), evaluate \(\nabla^2\phi\).

Use \(\nabla^2\phi=-k^2\phi\) for a free particle to find its energy.

What is the Hamiltonian operator for a nonrelativistic particle in potential \(U(\vec r)\)?

Why do boundary conditions matter in solving the three-dimensional Schrodinger equation?

A normalized wave function is multiplied by \(2\). What happens to its normalization integral?

If a wave function separates as \(X(x)Y(y)Z(z)\), what does that usually say about the potential?

Why is \(|\psi|^2dV\) a probability but \(|\psi|^2\) alone is a density?

What is the time-dependent Schrodinger equation in operator form?

Why must a bound-state wave function usually approach zero far from the binding region?

A particle is confined to a volume \(V\) with constant wave function magnitude. Estimate \(|\psi|\).

What is the main difference between the one-dimensional and three-dimensional time-independent Schrodinger equations?

Explain why the three-dimensional Schrodinger equation is linear.

What physical quantity is represented by an eigenvalue \(E\) in the time-independent equation?

Derive the time-independent equation by substituting \(\psi(\vec r,t)=\phi(\vec r)e^{-iEt/\hbar}\) into the time-dependent equation.