2. Vibrational Motion Harmonic motion occurs when a particle experiences a restoring force that is proportional to its displacement. F=-kx Where k is the force constant. The stiffer the spring, the greater the value of k. Force is also the gradient of the potential energy V. In 1-D:F=-dV/dx For F=-kx  V=1/2kx 2

3. Vibrational Motion For two masses connected by a spring: The force on particle 1 will be equal and opposite of the force on particle 2. This force will only depend on the relative distance between the particles. Define this relative distance as x = x 1 – x 2. If we are to speak in terms of this relative coordinate, then we must also use a relative mass to validate F = ma = -kx.

4. QM of the harmonic oscillator What is the SWE for a vibrational motion in terms of our relative coordinates x? Mathematicians solved this problem long before quantum mechanics. The solution has eigenvalues: Notice that E v is never zero. For v = 0, E v = 1/2h  This is defined as the zero point energy (ZPE) Without ZPE the uncertainty principle would be violated. We’d simultaneously know both the position (0 displacement) and momentum (0) of the particle. The energy level spacing is uniform:

5. The Harmonic Oscillator Energy level diagram The energy level diagram

6. The harmonic oscillator wavefunctions The wavefunctions (eigenfunction of the SWE) for a harmonic oscillator have the form:  (x) = N x (polynomial in x) x (bell-shaped Gaussian function)

7. The harmonic oscillator wavefunctions

8. The probability distributions Note: correspondence principle – the results are more classical as v increases

9. Properties of oscillators Show that  0 and  1 are orthogonal. Calculate the average displacement of a harmonic oscillator in its first vibrationally excited energy level (v = 1).

10. Vibrational Motion What are the turning points of a classical harmonic oscillator in its ground vibrational state (v = 0)? What is the probability of finding the particle outside the classical turning points? Tunneling – penetration through classically forbidden zones.

11. Vibrational Motion Show that the zero-point level of an harmonic oscillator is in accord with the uncertainty principle.

12. Useful Integrals

13. Rotational Motion in 2-D Consider a particle of mass m constrained to a circular path of radius r in the xy plane (A particle on a ring).

14. Rotational Motion in 2-D The de Broglie relation gives us the wavelength of a particle with momentum p. We must place boundary conditions on  such that condition (b) is met. The circumference of the ring must be an integer multiple of the wavelength

15. Rotational Motion in 2-D An acceptable wavefunction for this problem is: The probability density is independent of the angle so we know nothing about the particles location on the ring. The sign on m l indicates the direction of travel, just as the sign on e ikx indicated direction for our 1-D free particle. There’s also an angular momentum operator that can operate on Y to give the angular momentum of our particle.

16. Rotational Motion in 3-D A particle on a sphere must satisfy two cyclic boundary conditions; this requirement leads to two quantum numbers needed to specify its angular momentum state.

17. Spherical polar coordinates  We can discard any term than involves differentiating with respect to r since r is constant.

18. Rotational Motion in 3-D The SWE for rotational motion is: The wavefunctions are the spherical harmonics (see table 12.3). The energy levels are: There are 2l + 1 different wavefunctions (one for each value of m l ) that are degenerate. A level w/ quantum number l is (2l+1)-fold degenerate.

19. Angular Momentum for Particle on a sphere Since energy is quanitized it follows that angular momentum J should also be quantized. So the magnitude of the angular momentum = {l (l +1)} 1/2 ħ

20. Putting it all together Our particle on a ring in 2-D gave us the z-component of angular momentum = m l ħ where m l = l,l-1,…-l Think of m l as the angular momentum quantum number for the tip of a top recessing around the z-axis. The angular momentum of the spinning top will be specified by l.

21. Conclusion Quantum mechanics says that a rotating body may NOT take up any arbitrary orientation with respect to some specified axis. This orientation restraint is called space quantization. The quantum number m l is referred to as the orbital magnetic quantum number because it indicates the orientation of a magnetic field caused by the rotation of a charged body about an axis.

22. The Stern and Gerlach experiment verified the idea of space quantization. (a) A magnet provides an inhomogeneous field that Ag atoms must pass through. (b) Classically, the atoms should be deflected uniformly since they should have arbitrary configurations as they pass through the magnetic field. (c)The observed behavior agrees with space quantization imposed by QM. Space Quantization

23. Spin The angular momentum of a particle due to motion about its own axis is called spin. Spin of a particle is sophisticated and actually comes from the theory of relativity. DO NOT think of it as an actual spinning motion. For an electron, only one value of s is allowed, s = ½. There are thus 2s + 1 = 2 different orientations. m s = +1/2 (denotes as a or spin-up) m s = -1/2 (denoted as b or spin-down) Fermions are particles with half-integer spin (electrons & protons). Bosons are particles with integer spin (photons & neutrons).

24. Summary of Angular Momentum Quantum #’s NameSymbolRange of values Orbital angular momentum quantum numberl0, 1, 2, ….. Orbital magnetic quantum numbermlml 0,±1,…, ±l Spin angular momentum quantum numbers½ for an electron Spin magnetic quantum numbermsms ±1/2 for an electron

25. Methods of Approximation Every application we have encountered thus far has had an exact solution of the SWE. Most ‘real’ problems do NOT have exact solutions. There are two approximation methods for treating these ‘unsolvable’ problems 1. Perturbation theory 2. Variation theory

26. Perturbation Theory Assume Hamiltonian of our ‘unsolvable’ problem is a sum of: 1. A simple Hamiltonian, Ĥ (0), which has known eigenvalues and eigenfunctions. 2. A contribution Ĥ (1) which represents the extent to which the true Hamiltonian differs from the simple Hamiltonian.

27. Perturbation Theory Example Find the 1 st order correction to the ground-state energy for a particle in a box where the bottom of the box is well- shaped with a variation in the potential of the form V = -  sin(  x/L)

28. Variational Method Variation principle: If an arbitrary wavefunction is used to calculate the energy, the value calculated is never less than the true energy. The arbitrary wavefunction is the trial wavefunction. The variational method allows us to calculate an upper bound to the energy eigenvalue. We can include an adjustable parameter in our trial wavefunction and adjust the parameter to minimize the variational energy. The variational integral W ≥ E true