An important model quantum system: The harmonic oscillator

1. An important model quantum system: The harmonic oscillator What it is: an ideal spring, V(x)=(k/2) x2 Useful approximation for: -Vibration of a diatomic molecule -Vibrations of a polyatomic molecule of N atoms. Why are there 3N-6? -Vibrations of a crystal lattice In our discussions, x will be the displacement from the equilibrium position of the spring. ? F=-kx V(x) x=0

2. Newton’s equations ma=F for the 2 atoms: Let’s make it just one equation – easier to solve! Divide each equation by its mass and subtract “2nd equation – 1st equation: x2-x1=re+x if we define This is just Q: What is μ for an HCl molecule? (m1≈1amu, m2≈35 mu) For O2? (mO≈16 amu) The solution to this equation is a sine or cosine wave – show by substitution that the oscillation frequency is :

3. Kinetic Potential As always, the Hamiltonian is H = K + V = E. Making the usual classical→quantum substitutions p→-iℏ ∂/∂x, E→-iℏ ∂/∂t, we obtain the Schrödinger equation for our oscillator: Before we solve this differential equation, let’s use our ‘quantum intuition’ to think about what the solutions Ψ look like: -Ψ must satisfy the Heisenberg principle Particles at rest – not possible! E > 0 -At higher energy, Ψ must ‘wiggle’ more because K gets bigger, therefore p2 gets bigger, so d2Ψ/dx2 gets bigger. Remember: wavefunctions have a real (blue) and imaginary (red) part! To get the lowest possible energy wavefunction, how much should it ‘wiggle’?

4. Recall that only special wavefunctions have a definite energy E, those that satisfy Ψ(x,t) = Ψ(x) e-iEt/ℏ. Show it by applying the operator E→-iℏ ∂/∂t to this special Ψ. For these special “eigenstates,” the harmonic oscillator Schrödinger equation becomes (after dividing by what factor on both sides?): How do you do that? if we use units for X and P where μ=k=ℏ=1 for simplicity This equation has a unique property among Schrödinger equations! Recall that because x and p are “Fourier-conjugate”, if p→-iℏ ∂/∂x then also x→+iℏ ∂/∂p. If you substitute that into the Hamiltonian H = K + V = p2/2m + k/2 x2instead, you get Compare these two equations. Notice something? Ψ(x) therefore must equal Ψ(p). But recall that functions of Fourier-conjugate variables are Fourier transforms of one another. So Ψ(x) is its own Fourier transform. We saw earlier that only one type of function is its own F.T. What is it?

5. A: The Gaussian function! So Why is this one lowest? Hint: ‘wiggles’ The “0” subscript indicates the lowest energy wavefunction with quantum number n=0. Why 0 instead of 1, like for a particle in a box? Tradition! We could have called the lowest quantum number “1” and still make it all work out. Figure out what b and E0 (the lowest energy) are by plugging the Gaussian into the harmonic oscillator equation: Label n=0 on “the ladder:” Of course, there are eigenstatesΨn(x) with quantum numbers n>0 that ‘wiggle’ more, just like there were for the particle in a box. How do we get those, if we only have Ψ0(x)? Quantum energy levels are like a “ladder.” “Ladder operators” that take us up a step (excite the vibration of the molecule, as with a laser) or take us down a step (de-excite the molecule, like emission of a photon) will solve the equation and tell us what the energy levels and wavefunctions of a harmonically vibrating molecule look like.

6. The operators a and a† have the desired property: In the lecture notes we show that if we are in state |n> (n=0 would be our Gaussian), then a†|n> = √(n+1) |n+1> and a|n> = √n |n-1> Try this out on two examples. Let’s take and apply to it a† and a in our simplified units of μ=k=ℏ=1 for simplicity: What happened when you tried to descend the ladder from the n=0 ground state?

7. By plugging our wavefunctions for n=0, n=1, n=2 … thus obtained into the left-hand side of the Schrödinger equation we obtain on the right hand side the same eigenfunction, multiplied by the energy. The energy levels are where is the same frequency of the classical oscillator that we saw before The levels are EVENLY SPACED Compare to the particle-in-a-box spacing E~ In what ways are the wavefunctions similar to the behavior of classical particles? In what ways are they very different?

8. Classically Forbidden The ground state wavefunction is the most different from classical particle behavior Wavefunctions of higher energy squared, Pquantum(x) = |Ψ(x)|2, begin to resemble the classical probability Pclass(x). Wavepackets of higher energy resemble the classical motion of a particle, with a width Δx to satisfy Heisenberg’s principle. Web Link Local link ↑