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Expectation value of momentum

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Suppose a particle is in some state |psi1>. Then, the probability P that it is found to be in some (other) state |psi2> during a measurement is given by the square of the absolute value of the inner product:

P = |<psi2|psi1>|^2

The inner product can be expanded in the position basis as:

<psi2|psi1> = Integral over x of <psi2|x><x|psi1> dx

= Integral over x of <x|psi2>*<x|psi1> dx

= Integral over x of psi2(x)*psi1(x) dx, (1)

where the star denotes complex conjugation and we used that, in the position basis, the components of a state constitute, by definition, the wavefunction. Note that, for a continuum of states, like the possible positions a particle can be in, the square of the absolute value of the inner product yields a probability density, not a probability. E.g.:

|<x|psi>|^2 = |psi(x)|^2 is the probability ...

Solution Summary

We give a detailed proof of the fact that if the wavefunction is real valued, then the expectation value of the momentum is zero.

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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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