Section 2 Flashcards
h/2π (reduced Planck constant)
A wave that describes an electron should have
And p=mv
Simple, planewave
Energy of free electron in empty space
(Time dependent) Schrödinger equation
Simplest equation to describe free plane wave
Schrodinger’s equation of motion under a potential
-(e^2)/x
Quantum state of particle
Sqrt(2pi)
Normalisable, functions have
At a time t, uncertainty of position of plane wave approximated by
Heisenberg’s uncertainty, principle
Free Schrödinger equation can be used to determine
Wave function for all t starting with t=0 and
Probability density of a wave function is
Gaussian and centred at x-_0(t)
The centre of the probability density of a wave function moves with?
Velocity (classical velocity of particle)
For large times, how to deal with uncertainty of x
Ignore initial uncertainty and let
After a large time t, particles coordinates x_i wil most likely be in
(That p_1 should be p_i)
Linear operator for momentum
Let p^ _i act on the wave function of a plane wave
(As this wave a definite momentum p- ψ is just multiplied by the value of p_i
A wave function ψ is an eigenfunction function of Ο^ if ? What does that mean then?
With eigenvalue Ο€ R
This means that the observable described by Ο^ has a definite value in the state described by ψ
Expectation of Ο^ in a normalised state given
Where Ο^ is general observable
Rewrite Schrödinger equation using operators
- use momentum linear operator (63)
What does the Hamiltonian operator do
Generalises the classical expression below
What is the Hamiltonian operator (equation)
Hermitian operator
Prove momentum operator is Hermitian
Integration by parts to get 2nd last line
Compute time derivative of normalisation condition
Given that
the Hamiltonian (H^) is a Hermitian operator
And
Ψ_1 = Ψ_2 = Ψ
Therefore last line = 0
Therefore norm is constant
Conservation of norm is what?
As the norm differentiates wrt time to give zero
Norm is constant
Therefore if norm = 1 at one moment of time
It is 1 for all t
Total probability is conserved under time evolution <=> time evolution is unitary
What is the commutator
For any observable operator, the time derivative of its expectation value is given by
Prove result of time derivative of expectation of any observable operator
2.2 is definition of Hermitian
Sub in Hamiltonian for time derivative
Commutator of Hamiltonian operator and momentum operator
How to solve commutator of the Hamiltonian operator and the momentum operator
Compute commutator of potential of position and momentum operator
Where (63) is momentum operator = -ih(d/dx_i) (line through h)
Compute commutator of momentum from Hamiltonian and momentum operator
Combined solution to commutator of Hamiltonian operator and momentum operator
(87) is [H^, p^_i] = 1/(2m) [p^2, p^_i] + [U(x-), p^_i]
(91) is [U(x-), p^_i] = ihd(U(x-))/dx_i
Time derivative of momentum operator in any state
Using commutator of Hamiltonian (first line)
(2.3) formula for time derivative of expectation of any observable operator
[H^, x^_i] = ?
Time derivative of expectation of position operator
What are Ehrenfest Theorem?
Quantum contrast to classical equations of motion (classical below)
The Ehrenfest Theorem
Just top and bottom equations (middle used to derive bottom)
To integrate Gaussian function
Sqrt(π*σ)
Process when normalising wave
1) integrate abs val
2) let’s abs val equal 1
3) rearrange to isolate A
Normalising wave in R3
When changing variables to change dx3 to dx just cube entire integral
What is this
centre of Gaussian distribution of position
What is the wave function of a particle likely to be found at t=0 near x=x_0
What is an operator?
Linear map from vector space to self
O: V -> V
Prove time Unitarian
Using Schrödinger equation at top
3 facts about Hermitian operators
1) sum of Hermitian operators is Hermitian
2) product of Hermitian operators is Hermitian
3) all R is Hermitian
