Simple Quantum Mechanical Systems Flashcards
Time independent Schrödinger equation with Hamiltonian operator plugged in
Derive time independent Schrödinger equation with Hamiltonian plugged in
Factorise Ψ(x,t) for separation of variables
variable separated time independent Schrödinger equation? significance?
LHS Depends only on t, RHS depends only on x
Solution to LHS of variable separated time independent Schrödinger equation
How to form time independent Schrödinger equation with Hamiltonian plugged in
-factorise Ψ(x,t) = Φ(t)Ψ~(x)
-separate variables
(107 is RHS of variable separated time independent Schrödinger equations)
What does time independent Schrödinger equation with Hamiltonian plugged in suggest ? (What else does this mean)
Ψ~(x) is an EIGENFUNCTION of Hamiltonian Operator
A particle in state Ψ~(x) has definite energy value = E
Time dependent wave function
Where Ψ(x,t) = Φ(t)Ψ~(x)
General solution of time independent Schrödinger equation (particle in a box)
Ψ(x,t) is localised (t=0)
Time independent Schrödinger equation with non trivial potential takes what form w Hamiltonian operator
1d Particle in a box;
State assumptions
Particle moves along 0<x<L
Outside of this wave function vanishes
1d Particle in a box;
General solution to time independent Schrödinger equation with Hamiltonian plugged in
1d Particle in a box;
Determine Energy of particle
1d Particle in a box;
Normalise wave function
Given Ψ(x) = Acos(kx) +Bsin(kx)
=> |Ψ(x)|^2 = B^2(Sin(kx))^2
=>
1d Particle in a box;
Form time dependent solutions to Schrödinger equation
Then use to give corresponding probability densities
Where
E_n = (π^2h-2n^2)/(2mL^2)
gen solution is
Ψ(x) = Acos(kx) +Bsin(kx)
=> |Ψ(x)|^2 = B^2(Sin(kx))^2
And B = sqrt(2/L)
=>
Relate Ψ_loc to Ψ_n
Ψ_loc is a state that is a superposition of many eigenstates Ψ_n
1D particle in a box;
Decompose eigenstates of Ψ_loc (x)
1D particle in a box;
Decompose eigenstates of time dependent Schrödinger equation
(121) is solutions to time dependent Schrödinger equation
(124) is decomposition of localisation of time independent wave function
1D particle in a box;
Define a potential well
1D particle in a box, potential in a well;
Solve portion where potential is negative
What are assumptions ?
What is form of solution
1D particle in a box, potential in a well;
Even case
1D particle in a box, potential in a well;
Odd case
1D particle in a box, potential in a well;
Determine energy levels when -U_0 < E <
0
1D particle in a box, potential in a well;
What happens if E<= -U_0 ?
Derive uncertainty of a general operator Ο^^
I.e.
ΔΟ^2
And prove that this is equal to the expected squared derivation of Ο^^ from the expected value <Ο^^>
Ie
1D particle in a box, potential in a well, E > 0;
First define V_0
And what it means
V_0 = -U_0
For V_0 this is a potential barrier
If E < V_0 then the classical particle would get reflected
1D particle in a box, potential in a well, E > 0;
Define U(x) for 3 regions of potential well
1D particle in a box, potential in a well, E > 0;
Solve time independent Schrödinger equation for region 1
1D potential barrier, E > 0;
Solve time independent Schrödinger equation for region 2
1D particle in a box, potential barrier, E > 0;
Solve time independent Schrödinger equation for region 3
Same as region 1 with different coefficients
1D particle in a box, potential barrier, E > 0;
Full time dependent solution for a given p
Wave packet?
Called this why?
Ψ(x,t) for U(x) = 0
At t=0 localised near x=x_0 with uncertainty Δx and momentum localised around p
Oscillates in x, hence wave
Localised, hence packet
As time goes on, wave packet described by Ψ(x,t) moves to
Relate solution to wave packet to plane waves
p->q and the wave is no longer localised/idealised
Construct relation between free particle moving through potential barrier
Ψ(x,t)_scattering?
Choosing x_0 «_space;-L/2 so that Ψ(x,t) is localised entirely for x < -L/2 at t=0
Quantum tunelling?
Quantum particle can penetrate a potential barrier even if it has energy that is lower than the height of the barrier
1D particle in a box, potential in a well, E > 0;
Tunnelling probability is small if
Relate R(p) and T(p)
Add to equal 1
When is R(p) zero
When sin(p2L/h)=0
One deBroglie wavelength
h/p Where h is non reduced p.constant
1D particle in a box, potential in a well, E > 0;
U(x) has jump discontinuity where?
x= +-L/2
General solution to LHS of variable separated Schrödinger equation (particle in a box)
General solutions to RHS of Schrödinger equation (particle in a box)
Impose constraints if particle in box to find constants , e.g:
Ψ(0) = Ψ(L) = 0
What is the meaning of ψ localised
This is a approximation to a classical situation
The P density spikes in a small region and is almost zero everywhere else
Explain superposition
A particle can exist in many states
A localisation is the sum of these states that forms a wave packet
Scattering amplitudes?
A+ and C+
What is this?
What is their important property?
Coefficients of Fourier transform of Ψ(x,0)
They are Sharply peaked in abs value near q~p
which reflects the fact that Ψ(x,t) describes a particle whose momentum is approximately p with non zero uncertainty of P
When calculating Ψ(x,t)_scattering remember
We can replace -inf in integral bound with zero as area is negligible due to localisation
How to choose
It is chosen such that (below) is equal to Ψ(x,t) where Ψq(x,t) is the plane wave
Ψscattering(x,t) for region 1?
Ψscattering(x,t) for region 3?
Limitation to potential barrier
Modelled as instant jump, in practise would be a more shallow gradient
