Semester 2 Flashcards
If ψn(x) is a solution to the time independent schrodinger equation, give an expression for
Ψn(x,t) = ψn(x)f(t)
What is the physical meaning of the Hamiltonian?
It’s an operator corresponding to the total energy of the quantum system.
If you measure an observable Q on a particle in the state |ψ>, what are you certain to obtain?
One of the eigenvalues of the Hermitian operator Q.
How do you prove two vectors form an orthonormal basis?
Show that their inner product is equal to zero.
How is a quantum system represented in quantum mechanics?
By a wavefunction ψ(x,t)
What is the probability density of finding a quantum particle at position x?
P(x) = | ψ(x,t) |^2 dx
What does the normalisation condition allow you to determine?
The amplitude of the wavefunction because the integral of the absolute square of the wavefunction over all space must be equal to 1.
What is the average value known as in quantum mechanics?
The expectation value
What is wave mechanics?
The elementary formulation of quantum theory that is centred on the particles wave function
What is the evolution of the wave function ψ governed by in one dimensional space?
The Schrodinger equation
What is a useful way to simplify the schrodinger equation?
Looking for separable solutions, with everything dependent on t on one side and everything dependent on x on the other side.
How do you use separable solutions to simplify the schrodinger equation?
Substitute Ψ(x,t) for ψ(x)f(t)
What do solutions of the time independent schrodinger equation describe?
States of a particle with a definite energy known as stationary states.
What is “I”?
I = √-1
What is the wavelength defined as?
k = 2π/λ
What is the period defined as?
ω = 2π/T
where ω is the angular frequency
What is a time independent wave called?
A standing wave
What is a plane wave?
A wave that combines space and time oscillations to make a travelling wave f(x,t).
What is the intensity of a travelling wave f(x,t)?
|f|^2 = f*f = 1
What are the four dimensions of a plane wave?
- Real amplitude
- Imaginary amplitude
- x
- t
Why is a plane wave difficult to plot and how can this be solved?
Because it has four dimensions, to solve this you can make plots of planes for a given phase.
How do you switch between domains of time and frequency for a wavefunction?
Through fourier transforms
What are the three alternate ways of picturing the delta function?
- The limit of a sequence of functions.
- The derivative of the step function.
- The sifting functional.
What is meant by “free particle”?
A moving particle with no forces acting on it, in any region of space, so V (x) is constant for all of x.
Why is the energy of the free particle not quantised?
Because there are no restrictions on the value of k, which means there are no restrictions on the value of E.
What do the first and second terms of the free particle represent?
The first term represents a plane wave travelling to the right.
The second term represents a plane wave travelling to the left.
Why can a free particle not exist in a stationary state?
Because the integral of the wavefunction diverges meaning it cant be normalised. This means there is no such thing as a free particle with a definite energy.
What is the De Broglie rule?
λ = 2π/k = h/p
What is the Einstein relation for energy?
E = hc/λ = hω (wave aspect)
What is the Einstein relation for a free non-relativistic particle?
E = p^2/2m (particle aspect)
What is a wave packet?
A wave packet can be thought of as a wave which is modulated by an envelope function.
What material can be used to create a quantum well?
A semiconductor heterostructure such as GaAs or AlGaAs
How can you observe transitions between energy levels in a quantum well?
Optical spectroscopy
What is the current for a stationary state where Ψ(x) is purely real?
0
What is the superposition of states needed for?
To carry a current
What do fixed boundary conditions in quantum mechanics result in?
Discrete energy levels and wave functions which are standing waves and carry no current
What are Born-Von Karman boundary conditions?
Periodic boundary conditions that involve repeating the system periodically with the same wave function in each system.
What are the allowed values of k for the normalised Born Von-Karman wavefunction?
kn = 2πn/L
where n must be an integer
In the normalised Born Von-Karman wavefunction, what does each value of k correspond to?
A plane wave and a discrete electron wavefunction with a characteristic wavefunction.
How can you count the number of electron states in a material?
Through knowing the separation of states in k-space
What is the density of states function used for?
To describe quantum systems
Why is the density of states function g(E) used to describe quantum systems?
Because a complete description of a system requires the energies and wavefunctions of all its states, and this is impossible for all but the simplest physical systems.
What is the definition of the density of states function?
g(E)δ(E) is the number of states in the system whose energies lie in the range of E to E + δE
