Quantum Mechanics 1

what type of equation is TDSE? What does this mean?

Linear differntial equation: Superposition principal holds if ψ1 and ψ2 are solutions, then Aψ1 + Bψ2 is also a solution

What’s the difference between the TDSE and the TISE, and when do you use each one?

TDSE (time-dependent Schrödinger equation)

• It’s the main equation that tells you how a wave function changes over time.

• Energy eigenfunctions (with definite energy) as well as superposition states, satisfy the TDSE.

• Superpositions work because the equation is linear. If you add two solutions together, you get another solution.

• It’s first order in time. That means if you know the wave function at one moment, you know it at every later moment. This only holds as long as nobody makes a measurement.

TISE (time-independent Schrödinger equation)

• It’s an energy eigenvalue equation. Only states with one definite energy solve it. Superpositions don’t.

• You get it by splitting the wave function into a “space part” times a “time part” (separation of variables). That only works because the potential energy doesn’t change with time.

• The time part is the same for every system. The spatial part depends on the potential energy, so each system has its own TISE and its own solutions.

Conditions a valid wave function must satisfy, and give the reason for each.

• Normalization condition: To allow for a probability interpretation

• ∂ψ/∂x must be continuous except at points where the potential energy is infinite.

• Square integrable: Otherwise, the total area under |ψ|² is infinite and you can’t normalise it.

• Function continous: no jumps, probaility woould be ambigous

What does an expectation value mesure physically? How is it different from the result of a single measurement?

The average of a set of measured values of identical replicas whose initial states are all identical. And we do this because measuring the particle usually alters it

What does the uncertainty of an observable represent?

It’s not a flaw in measuring equippment,It’s the spread (standard deviation) of the possible measurement results, based on the state before you measure.
In quantum mechanics there’s no definite value to measure.

What does probability current describe physicsally

Probability per time past a unit point.

Probability around all space is always 1, the currrent tracks how it moves around

Why are the energies discrete in the infinite well but continuous for a free particle?

The walls have infinite potential energy, so the wave function must be zero at both walls. That means there are only certain wavelengths allowed. Each allowed wavelength gives one energy, so the energies are discrete (n = 1, 2, 3, …).

Free particle: there are no walls, so no boundary conditions. Any wavelength is allowed, so any positive energy is allowed (continuous).

Why can't the ground state energy be zero?

Because if E = 0 that would lead to ψ0(x, t) = 0 (it would be a straight line) ,implying there is not particle in the well, so you wouldn’t be able to normalize it.

Why do the eigenfunctions oscillate faster as n increases? What's the link to kinetic energy?

• The average kinetic energy of a wave function depends on how steep it is. More humps in the same width at the same height means steeper slopes, which means more kinetic energy.”

• Inside the well the potential energy is zero, so all the energy is kinetic. The average kinetic energy of the nth state is just its energy, Eₙ.

• Higher n means higher energy, so more kinetic energy, so more wiggly. That means a shorter wavelength and more nodes.

• Cross-check (de Broglie): a shorter wavelength means a bigger momentum, which means more kinetic energy.

Explain in words what it means for the energy eigenfunctions to be orthonormal, and what it means for them to be complete.

Orthonormal means that the eigenfunction is normalized and orthogonal:

• Normalised: total probability = 1

• Orthogonal: Any two different eigenfunctions have zero overlap

Complete: any valid wave function for the system can be written as a combination (superposition) of the energy eigenfunctions. you can always expand a state and so work out its time evolution.

State the Bohr correspondence principle. In what sense does it hold for the infinite well, and in what sense doesn't it?

For high quantum numbers, quantum results reduce to classical ones.

• In the classical well: the particle bounces back and forth at constant speed, so every position is equally likely (probability density is flat)

• For quantum the probability density is NEVER flat, it keeps oscillating

• Over an interval, the oscillations average out so the probability in that interval tends to the classical value

What does an expansion coefficient tell you? When is it large, and when is it negative?

|cn|² = Prob of obtaining the energy En if a measurement of energy is taken.

If its big, higher probability and viceversa

The coefficient is the overalp between the state and the nth eigenfunction. (multiplying to functions together and taking area as result)

• large and positive: state and nth eigenfunctions —> similar shapes

• large and negative: state is an upside down version of eigenfunction

• small or zero: shapes dont match

What are the two types of time evolution in quantum mechanics?

1. Deterministic time evolution, with no measurment: the wave function evolves smoothly according to TDSE

2. Non-Deterministic time evolution, with a measurment: wave function collapses onto one of the eigenfunctions of the thing measured. You only know probabilities

What happens to the wave function when you measure energy? How does it evolve afterwards?

The wave function collapses to just the nth energy eigenfunction. It’s still a wave function, jus a single one now, with the other contributions gone

Afterwards: It stays in that state. Only an overall phase changes, so its probability density is constant in time

If you measure an observable whose operator doesn't commute with the Hamiltonian, how does the state evolve after the measurement?

The state collapses to an eigenfunction of that observable, but that eigenfunction isn’t an energy eigenfunction. It’s a superposition of several energies.

Measuring the same thing again later could give a different result.

if the operator commutes with the Hamiltonian, the collapsed state is also an energy eigenfunction, so it stays put.

Why can't a plane wave be normalised? What does that tell you about the particle's position?

A plane wave is a complex exponential, so its size is the same everywhere. Its probability density is constant in all space. So wehn you add it up over all space (-inf, inf) = INF

Position: The particle is equally likely to be anywhere, so its psoition is compltely undefined

To build a localised free particle, why do you need to integrate over momenta rather than add up a discrete set of them?

• Particle isn’t loclaised (flat prob density)

• A discrete sum: probability has peaks —> still not localised and still can’t be normalised

• Integral over momenta: single bump —> localised and normalisable: a wave packet

• Particle no longer has definite momentum

In a Gaussian wave packet, what is the physical meaning of the width of the momentum (or wave number) distribution?

The width is the uncertainty of the wavenumber: it’s the measure of the range of wavenumbers of the component plane waves that make up the packet

What does “dispersive” mean, and why are free-particle matter waves dispersive?

Dispersive means different wavelengths travel with different speeds

Free particle matter waves are dispersive because the dispersion relation w(k) is not proportional to k.

For a free-particle wave packet, what’s the difference between phase velocity and group velocity, and which one describes how the packet moves?

Phase velocity: the speed of a single plane wave with one wave number, it depends on the wave number.

Group velocity: the speed of the packet’s peak. It only exists for a packet of many wave numbers. (Centred at k0)

Why does a free-particle wave packet spread out over time?

• packet is a mix of different wavelengths

• matter waves are dispersive, so shorter wavelegth travel faster

• the fastt parts pull ahead and the slow parts fall behind, so the packet gets wider

Does the momentum uncertainty of a free-particle wave packet change with time? Why or why not, and is this true for every system?

No, it does not, each momentum component only gains a phase, so the momentum distribution stays the same.

Not a general result, in the presence of external forces, Δ𝑝 will in general depend on time

What does the Ehrenfest theorem say, and how does a free-particle Gaussian wave packet illustrate it?

The Ehrenfest theorem says that expectation values follow classical physics.

Gaussian packet: its peak (the average position) moves at a constant velocity, the group velocity. That’s exactly what a classical free particle does with no forces.

How can you tell whether a given wave function (or momentum wave function) describes a state of definite momentum?

• Momentum distribuition is a definitely narrow spike at one value

• Position state it’s a plane wave with the same prob density everwhere (undefined)

• Operator test: apply the momentum operator. If you get back a constant times the same function, it’s definite, and the constant is the momentum.

How is the energy spectrum of the 1D harmonic oscillator different from that of the infinite square well, and why is the oscillator’s ground state energy non-zero?

• Oscillator: Equally spaced energies starting at n=0, with ground state energy half a quantum

• Ifinite well: gaps grow with nm starting at n=1

• Ground state isnt zero: being still at the bottom would mean definite position and momentum at once, which breaks heisenberg.

In words, how are the energy eigenvalues and eigenfunctions of the 3D harmonic oscillator related to those of the 1D oscillator?

The 3D oscillation splits into three independent 1D oscillators (x,y,z), each with its own quantum number starting at 0

Energies add: The total is the sum of the three 1D oscillator

Eigenfunctions multiplt: each is the product of three 1D eigenfuntions

What is degeneracy, and why does it appear for the 3D harmonic oscillator but not the 1D one?

Degeneracy: differnt eigenfunctions with the same eigenvalue. Here different states with the same energy

3D: Energy depends only on the sum of 3 quantum numbers, differnt combinations give same energy (ground state is not degenerate)

1D; only one quantum number, so one state per energy

Why does the orthogonality of a Hermitian operator’s eigenfunctions matter physically?

Hermitian operators have orthogonal eigenfunctions for differnt eiggenvalues.

Physically, differnt measurments outcomes are mutually exclusive. A state with one definite value has zero chance of giving a differnt one.

What does it mean physically if two operators commute, and what does it mean if they don’t?

• Commute means compatible: they share eigenfunctions, so a state can have definite values of both. Measuring one doesn’t disturb the other, and the order doesn’t matter.

• Don’t commute means incompatible: no shared eigenfunctions, so you can’t have definite values of both. Measuring one puts the other into a superposition. The commutator sets the minimum product of the uncertainties.

Difference between expectation value and most likely interval to find the particle?

The expectation value is the average value of position one obtains when one measures the position of a large number of identical quantum systems all described by this wave function. This is the difference between the mean and the mode of the probability density.