Topic 8 Flashcards
What is a linear harmonic oscillator?
A system which is any physical environment where the system can reach a stable equilibrium position at a minimum with energy and which then undergoes oscillations or vibrations about the minimum of the energy potential surface.
What can the linear harmonic oscillator be approximated as?
The harmonic oscillator can be approximated to the bottom of a generic potential well, so we can take any potential well which has a minimum.
For sufficiently small displacements from the minimum of that, we can model the potential as a linear harmonic oscillator.
What is the quantum linear harmonic oscillator?
The potential is the same as the classical LHO
TISE tell us the spatial part of the wave function psi of x.
We get energy from the TISE. Each energy eigenstate is oscillating in time.
What is the TISE form for the QLHO?
−ћ/2mψ’‘(x) + 1/2mω^2x^2ψ(x) = Eψ(x)
What is the normalized-to-unity total energy eigenfunctions equation?
ψ_n(x) =(1/n!2^na√π)^1/2
H_n(x/a)e^−x^2/2a^2
a = sqrt(ћ/mω)
What is the Hermite polynomial?
A special class of functions which are solutions to the energy eigenfunctions equation
What are the corresponding energy levels to the energy eigenfunctions equation?
E_n = ћω(n +1/2)
The energy levels are labelled by the quantum number n
n = 0, 1, 2…
0 is also included this time
What are the charactertics of the wavefunction?
The wavefunctions ψn(x) have even
parity for n = 0, 2, 4, . . . and odd
parity for n = 1, 3, 5, . .
What is the Correspondence Principle?
The behaviour of quantum mechanical systems should
approach that of the classical system in the limit of large quantum numbers
What is the implication of the quantum LHO?
The Correspondence Principle
How else can we interpret the solution to the quantum LHO?
Using ladder operators
Aˆ ≡ sqrt(mω/2ћ)(xˆ +i/mωpˆ)
and Aˆ† ≡ sqrt(mω/2ћ)(xˆ - i/mωpˆ)
What are the chracteertics of ladder operators?
They not Hermitian operators and do not correspond to physical observable
What is a lowering operator?
Acting with Aˆ reduces
n by 1: Aψˆ
n(x) = ψn−1(x) (except if n = 0, in which case Aψˆ
n(x) = 0)
What is a raising operator?
Acting with Aˆ† on an eigenstate increases n by 1: Aˆ†ψn(x) = ψn+1(x). Similarly
What is the interpretation of ladder operators?
Aˆ†Aψˆ
n(x) = nψn(x) and so n is the eigenvalue of of Nˆ = Aˆ†A
