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
State the momentum operator.
Check
State the total energy operator.
Check
State the time operator.
Check
State the position operator.
Check
State the potential energy operator.
Check.
Derive the KE operator.
Check.
Derive the angular momentum operator.
Check
What happens when you use the energy operator on a free particle wave function.
Check
The result is Plank’s formula
What happens when you use the momentum operator on a free particle?
Check
The result is the de Broglie formula
Derive the Heisenburg Uncertainty Principle using the commutation of operators.
Check derivation.
Solve the time dependent Schrödinger equation
Check
Solve the time independent Schrödinger equation
Check
Find the total wavefunction putting together the time dependent solution and the spatial dependent solution.
Check.
Normalise the wave function for a particle in a 1D box.
Check
What is Hilbert space?
An infinite dimensional space that can be defined by eigenfunctions that form a complete orthonormal set (used as basis vectors).
Even if the wavefunction is not an eigenfunction of the observable, what is the outcome always?
An eigenvalue.
If a measurement is performed of any observable Q, and the eigenvalue q is obtained, what happens to the system?
The subsequent state of the system will be the eigenfunction u(r).
What is non-locality?
Where a measurement at one location can affect the result at a subsequent without any signal passing between the two due to quantum entanglement.
Derive the wavefunction for a particle of energy E incident on a potential step V.
Check derivation.
Derive the wavefunction for a particle of energy E incident on a potential step V and V>E.
Check derivation.
What is quantum tunnelling?
The phenomena where there is a finite probability of finding a particle in a forbidden region (potential barrier), and a finite probability that a particle will pass through a finite width barrier even though the potential is higher than the total energy.
How does the probability of quantum tunnelling vary with width of the barrier?
Falls exponentially with increasing width.
3 examples of quantum tunnelling.
STM
Alpha decay
Covalent bonding.
How does quantum tunnelling work in STM?
The small electrical bias applied to the tip and the sample becoming a tunnelling gap.
How does quantum tunnelling work in alpha decay?
An alpha particle must tunnel through a barrier formed by the short range attraction to other nucleons and the longer range electrostatic repulsion by the remaining protons.
How does quantum tunnelling work in covalent bonding?
If the height of the potential barrier is not infinite, but greater than the total energy, then evanescent waves exist outside the well.
State the Schödinger equation in 3D Cartesian coordinates.
Check
State the Schödinger equation in 3D polar coordinates.
Check
State the solution for the energy of a particle in a 3D box.
Check
What is degeneracy?
When states with different quantum numbers have the same energy they are described as degenerate.
If an energy state has n number of degenerate states, it has n-fold degeneracy.
Show that x, y and z components of angular momentum do not commute.
Check
Show that angular momentum squared and the z component of angular momentum commute.
Check
Derive equations showing the spherical harmonics are eigenfunctions of the angular momentum squared operator and z component of the angular momentum operator.
Check derivation.
What are the allowed values of the m quantum number?
-l ≤ m ≤ +l
Solve the radial energy equation for the energy of a hydrogen atom.
Check
Calculate the most probable radius of an electron in a hydrogen 1s orbital.
Check
What is the eigenvalue of spin?
h(bar)^2s(s+1)
What values of spin do electrons always take?
±1/2
What values of spin do bosons always take?
Integer spins.
State the Pauli exclusion principle.
No two indistinguishable fermions can occupy the same quantum state.
Why do lower n valued orbitals tend to fill first?
Lower n values are more bound so fill first. For a given principle quantum number n, the larger l tend to have a larger mean radius and are more screened. This lifts the degeneracy such that lower l values are more bound than the high l vales within the same n shell so fill first.
Hund’s first rule.
The total spin S is the sum of the ms values for individual electrons. This is consistent with the Pauli exclusion principle, the electrons will fill to maximise S.
Hund’s second rule.
The total orbital angular momentum L, is the sum of the ml values for the individual electrons. This is consistent with the Pauli exclusion principle, the electrons will maximise L.
Hund’s third rule.
If the partially filled subshell is not more than half full, then the total angular momentum is h(bar)J where J = |L-S|, and if it is more than half full then J = L+S.
Calculate the total angular momentum of a silicon atom and write down its term symbol.
Check
For H2, using linear combinations of orbitals, show that cA = cB for bonding and cA = -cB for antibonding.
Check.
