7.3 The Schrӧdinger Equation

A particle with mass m is moving along the x-axis in a potential given by the potential energy function $U \left(x\right) = 0.5 m \omega^{2} x^{2}$. Compute the product $\Psi \left(x , t\right)^{\star} U \left(x\right) \Psi \left(x , t\right) .$ Express your answer in terms of the time-independent wave function, $\psi \left(x\right) .$

If a quantum particle is in a stationary state, does it mean that it does not move?

Suppose a wave function is discontinuous at some point. Can this function represent a quantum state of some physical particle? Why? Why not?

Combine the equation and the equation to show $k^{2} = \frac{\omega^{2}}{c^{2}} .$

Show that $\Psi \left(x , t\right) = A \text{sin} \left(k x - \omega t\right)$ and $\Psi \left(x , t\right) = A \text{cos} \left(k x - \omega t\right)$ do not obey Schrӧdinger’s time-dependent equation.

A particle with mass m is described by the following wave function: $\psi \left(x\right) = A \text{cos} k x + B \text{sin} k x$, where A, B, and k are constants. Assuming that the particle is free, show that this function is the solution of the stationary Schrӧdinger equation for this particle and find the energy that the particle has in this state.

Find the expectation value of the square of the momentum squared for the particle in the state, $\Psi \left(x , t\right) = A e^{i \left(k x - \omega t\right)}$. What conclusion can you draw from your solution?