Please see attached file for full problem statement.© BrainMass Inc. brainmass.com October 24, 2018, 11:43 pm ad1c9bdddf
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 ...
We give a detailed proof of the fact that if the wavefunction is real valued, then the expectation value of the momentum is zero.
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.View Full Posting Details