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.
