Wave mechanics and Quantum Flashcards
Describe what would happen classically in the photoelectric effect experiment.
Electrons should be emitted after some delay, with an energy dependent on the intensity of light.
Describe what happens experimentally in the photoelectric effect experiment.
Electrons are always emitted instantaneously, even at low intensities, with energies independent of that intensity but dependent on the wavelength. No electrons are emitted below a certain wavelength.
Plank equation for energy.
E=hf
Describe what classically happens for spectral lines.
Electrons should not be stable in their orbits but should lose energy and spiral in towards the positively charged nucleus.
Describe what experimentally happens for spectral lines.
The arrangement of electrons and nucleus is stable, with electrons occupying distinct energy levels, as shown by lines in spectra from discharge lamps.
Equation linking angular momenta of electrons to quantised units.
mvr = lh(bar)
Derive the energy level model for a H atom and the Rydberg.
Check derivation.
de Broglie relation.
p = h(bar)k = h/λ
§What is a wavepacket?
Combination of a range of wavelengths.
From the idea of a wavepacket, simply derive a version of Heisenburg’s Uncertainty Principle.
Check derivation.
Calculate the total energy of a hydrogen atom from KE and PE.
Check…answer should be a Rydberg.
Show how measuring position and momentum of an electron accelerated through a well defined potential does not break the uncertainty principle.
Check derivation.
Sum two travelling waves to make a standing wave.
Check
Why can’t standing waves be used to describe the electron?
Antinodes mean places where electrons cannot exist.
Why can’t a travelling wave be used to describe the electron?
Can’t exist over all space.
What kind of wave functions do we use for electrons?
Complex.
Born probability of a travelling wave function.
P(x,t) = |ψ(x,t)|^2 = ψ*.ψ = A^2
Derive the time independent Schrödinger Equation in 1D
Check derivation.
Under fixed boundary conditions, what happens when a travelling wave reaches the boundary?
It reflects and changes sign.
Under free boundary conditions, what happens to a travelling wave when it reaches a boundary?
It reflects without a change in sign.
Derive an expression for phase velocity.
Check
State expression for group velocity.
Check
Find both the phase and group velocities for a photon.
Check
Find both the phase and group velocities for an electron.
Check.
The first postulate of quantum mechanics.
For any particle moving in an external potential, there is an associated wave which is a single valued function of the spatial coordinates and of time. The wave function determines everything that can be known about the particle.
The second postulate of quantum mechanics.
For every physical observable, Q (energy, momentum etc), there is an associated mathematical operator Qˆ. A measurement of the observable, Q, gives a result which is one of the eigenvalues of the eigenvalue equation for the operator.
Qˆu(r) = qu(r)
q is eigenvalue.
The third postulate of quantum mechanics.
The functions u(r) are eigenfunctions of Q and form a complete orthonormal set so that any arbitrary normalised wave function ψ(r) can be expressed as a linear combination of the eigenfunctions.
The fourth postulate of quantum mechanics.
If a particle has a wave function ψ(r), which may or may not be an eigenfunction of the observable Q, then the probability that a measurement of Q will result in the eigenvalue q is given by:
P(q) = (∫ u*(r)ψ(r)dV)^2
The fifth postulate of quantum mechanics.
The measurement of an eigenvalue, q, immediately changes the wave function of the particle from ψ(r) to u(r).
The sixth postulate of quantum mechanics.
The mean value (aka expectation value) of the observable, Q, for particles with wavefunction, ψ(r), will
= ∫ψ*(r)Qˆψ(r)dV
