harmonic oscillator

What distinguishes the harmonic oscillator from the infinite potential well

The potential is finite for all finite xxx and diverges only as x goes to + or - infinity

What is the Hamiltonian of the quantum harmonic oscillator

H^ = P²^/2m + 1/2mw²X²^ where w = ROOT(k/m)

What equation defines the energy eigenstates

H^/n> = En/n>

What is the time-independent Schrödinger equation in position space

[-h(dash)²/2m d²/dx² + 1/2mw²x²]un​(x)=En​un​(x)

What are the energy eigenvalues of the quantum harmonic oscillator

En = )n + 1/2)h(dash)w

What is the ground state energy

E0 = 1/2h(dash)w

Why is the ground state energy nonzero

Due to quantum fluctuations and the uncertainty principle.

What is the general form of the harmonic oscillator eigenfunctions

un(x) = Cne^-alpha²x²/2 Hn(alphax)

what is alpha

alpha² = mw/h(dash)

What are Hermite polynomials

polynomials Hn​ of degree n appearing in the oscillator eigenfunctions

State the orthogonality relation of Hermite polynomials

integral from infinity to -infinity of e^-s² Hm(s)Hn(s)ds = 2^nROOT(pi)n!deltamn

What symmetry does the harmonic oscillator potential have

V(x) = V(-x)

What is the parity of the nth eigenstate

P^un(x) =(-1)^nun(x)

How do eigenfunctions alternate in parity

even n is even parity and odd n is odd parity

What is the expectation value of position in a stationary state

<x>n = 0 as the probability density is symmetric under x going to -x

What is the mean square displacement in state n

,x².n = n +1/2/alpha²

How is energy related to position variance

En = mw²<x²>n

what does /un(x)/² represent

The probability density of finding the oscillator at position x

What dimensionless operators are introduced

4^ = ROOT(mw/2h(dash))X^, n^ = P^/ROOT(2mh(dash)w)

What is the Hamiltonian in terms of $^ and n^

H^ = h(dash)w(n²^ + $²^)

What is the commutation relation between ξ^\hat\xiξ^​ and η^\hat\etaη^

[$^ , n^] = i/2

How are annihilation and creation operators defined

a^ = $^ +in^ and a^cross = $^ - in^

express a^ in terms of X^ and P^

a^ = ROOT(mw/2h(dash))W^ + i/ROOT(2mh(dash)w)P^

What is the key physical role of a^ and a^cross

they lower and raise the energy by one quantum h(dash)w