Write the time-dependent Schrodinger equation for a nonrelativistic particle in one dimension with potential \(V(x)\).

Identify the Hamiltonian operator for a one-dimensional particle in potential \(V(x)\).

For a stationary state \(\psi(x,t)=\phi(x)e^{-iEt/\hbar}\), what happens to \(|\psi|^2\) in time?

Derive the time-independent Schrodinger equation from \(\psi(x,t)=\phi(x)e^{-iEt/\hbar}\) for a time-independent potential.

Show that \(\psi=Ae^{i(kx-\omega t)}\) is a free-particle solution only if \(\hbar\omega=\hbar^2k^2/(2m)\).

In a region where \(V\) is constant and \(E>V\), what is the spatial form of a stationary solution?

In a region where \(V\) is constant and \(E<V\), what is the spatial form of a stationary solution?

What matching conditions apply to \(\phi\) at a finite step in potential energy?

Why may \(d\phi/dx\) be discontinuous at an infinite potential wall?

A proposed stationary state in a finite potential is continuous but has a kink where \(V\) is finite. Explain why it cannot satisfy the Schrodinger equation there.

For a constant potential \(V_0\), express the wave number \(k\) of an oscillatory stationary state in terms of \(E\), \(V_0\), \(m\), and \(\hbar\).

If \(\phi_1\) and \(\phi_2\) are energy eigenstates with energies \(E_1\) and \(E_2\), write the correct time evolution of \(c_1\phi_1+c_2\phi_2\).

A normalized state is \(\psi=c_1\phi_1+c_2\phi_2\), where \(\phi_1\) and \(\phi_2\) are orthonormal energy eigenstates. Find the expectation value of energy.

Check the dimensions of the kinetic-energy operator \(-\hbar^2(2m)^{-1}d^2/dx^2\).

Why is the Schrodinger equation linear important for superposition?

A real stationary bound-state wave function in one dimension has probability current \(j=(\hbar/m)\mathrm{Im}(\phi^*d\phi/dx)\). What is \(j\), and why?

Use integration across a finite potential step to justify derivative continuity for \(\phi\).

A wave function is an energy eigenstate of a time-independent Hamiltonian. Show that the probability density is stationary even though the wave function is not constant in time.

For a constant potential region, explain how the same differential equation predicts both propagation and evanescence.

Prove that two normalizable eigenstates of a Hermitian Hamiltonian with distinct energies are orthogonal.