# Introduction to quantum mechanics past paper

2. Two possible wave functions for states of a particle, with definite energies E_1 and E_2 are: see attachement for equations.

- Explain why these are called stationary states.

- Write down a wavefunction for a non-stationary state for which the expectation value of the energy is (1/3*E_1) + 2/3*E_2).

- Show that the probability density for position for this state oscillates with time.

- Calculate the frequency of the oscillations if E_1 = 1.2eV and E_2 = 0.3 eV.

- From a semi-classical perspective, what would be the consequences of such oscillations if the particle were electrically charged?

- The expectation value of the energy-squared for this state is: see attached. Calculate the uncertainty in the energy.

3. A particle of mass m moves in a two-dimensional simple harmonic oscillator potential of the form (see attached), where W_o is a constant. Write down the two-dimensional version of the time-independent Schrodinger equation for this situation.

- Consider a stationary-state solution of the time dependent equation of the form (see attached), where n_x and n_y are quantum numbers. What is the energy of this state (see attached), in terms of n_z, n_y, and W_o?

- What is the degeneracy of the state with energy (see attached).

- See attached for second last question.

- Describe the energy and angular momentum properties of the new wavefunctions.

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The solution provides answers to a past paper on an introduction to quantum mechanics. The solution discusses wavefunctions, stationary and non-stationary states, oscillating subjects, etc.