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Vibrational Motion

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 .

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Vibrational Motion

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  1. 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=1/2kx2

  2. 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 = x1 – x2. • If we are to speak in terms of this relative coordinate, then we must also use a relative mass to validate F = ma = -kx.

  3. 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 Ev is never zero. For v = 0, Ev= 1/2hn. • 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:

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

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

  6. The harmonic oscillator wavefunctions

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

  8. Properties of oscillators • Show that Y0 and Y1 are orthogonal. • Calculate the average displacement of a harmonic oscillator in its first vibrationally excited energy level (v = 1).

  9. 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.

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

  11. Useful Integrals

  12. 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).

  13. 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 Y such that condition (b) is met. • The circumference of the ring must be an integer multiple of the wavelength

  14. 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 ml indicates the direction of travel, just as the sign on eikx 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.

  15. 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.

  16. Spherical polar coordinates Y(q,f) = Q(q)F(f) • We can discard any term than involves differentiating with respect to r since r is constant.

  17. 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 ml) that are degenerate. • A level w/ quantum number l is (2l+1)-fold degenerate.

  18. 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)}1/2ħ

  19. Putting it all together • Our particle on a ring in 2-D gave us the z-component of angular momentum = mlħ where ml = l,l-1,…-l • Think of ml 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.

  20. 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 ml 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.

  21. Space Quantization • 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.

  22. 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. • ms = +1/2 (denotes as a or spin-up) • ms = -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).

  23. Summary of Angular Momentum Quantum #’s

  24. 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 • Perturbation theory • Variation theory

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

  26. Perturbation Theory Example • Find the 1st 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 = -e sin(px/L)

  27. 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 ≥ Etrue

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