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Lecture 4: in a 2D box, degeneracy, harmonic oscillator

1 in a 2D Box

In this case, the potential is given by V (x, y)=0 0 ≤ x ≤ a, 0 ≤ y ≤ b = ∞ otherwise The Hamiltonian operator is given by ~2 d2 d2 − + + V (x, y) 2m dx2 dy2  and the corresponding Schr¨odinger equation is given by ~2 d2ψ(x, y) d2ψ(x, y) − + = Eψ(x, y) 2m  dx2 dy2  inside the box and ψ(x, y) = 0 outside the box. The wavefunction inside the box can be solved by separation of variables ψ(x, y) = ψx(x)ψy(y) which can be shown to lead to the equations ~2 d2ψ (x) − x = E ψ (x) 2m dx2 x x and ~2 d2ψ (y) − y = E ψ (y) 2m dy2 y y This is simply the product of a two independent Particles in 1-D boxes so we have 2 n πx ψ (x) = sin( x ) x ra a

1 and 2 n πy ψ (y) = sin( y ) y r b b where n2 h2 E = x nx 8ma2 and n2h2 E = y ny 8mb2

The total energy E is now quantized by two numbers, nx and ny and is given by h2 n2 n2 E = x + y nx,ny 8m  a2 b2 

The lowest energy state is nx = ny = 1. If a > b, the next lowest energy state is nx = 2,ny = 1. When a = b, we have a degeneracy Enx,ny = Eny,nx. We can extend this problem to the following situations:

1. Particle in a 3D box - this has many more degeneracies. This is the classic way of studying density of states in metals or other free- systems.

2. 2 or more noninteracting particles in a box. In this case, we have to worry as to whether the particles are fermions or bosons and keep track of Pauli’s exclusion principle if they are fermions. This is particularly relevant for the modeling of pi- in conjugated alkenes as par- ticles in a box.

3. Particles in boxes that are shaped differently, (Use of curvilinear coordinates).

Many of the important principles of are illustrated by the various particle-in-a-box systems that we have studied. Quantization, degeneracies, role of dimensionality, etc. are naturally seen in these systems. We now move to other quantum systems wherein different aspects of the problems relating to atomic and molecular structure are seen.

2 2 The 1-D Harmonic Oscillator model

We have considered the particle in a box system which has either V (x) = 0 or V (x) = ∞. We now want to consider a system where V (x) is a quadratic function of x given by 1 V (x) = kx2 2 The motivation for this choice is the classical harmonic oscillator problem of U(x)

x a and a spring system shown below where x is the displacement from the equilibrium position. Interestingly, this can also be viewed as a 2 particle system interacting with a spring . When the equations of motion are converted to center of mass frame, the relative motion of the two particles follows the harmonic motion described above. For this reason, this turns out to be an important starting model for describing vibrations in diatomic molecules. The Hamiltonian operator for this system is given by ~2 d2 1 H(x) = − + kx2 2m dx2 2

3 leq

x m

The value of x is assumed to vary from −∞ to +∞. This might seem a little surprising at first since the spring cannot be compressed more than its equilibrium length. However, we keep in mind that the harmonic motion is only for small displacements, so there is a separation of length scales. In other words, we are assuming that x<

~2 d2ψ(x) 1 − + kx2ψ(x) = Eψ(x) 2m dx2 2 This rather simple looking equation turns out to be fairly difficult to solve. We will not try to derive the solution here, but simply take the solutions and examine the results.

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