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Unformatted text preview: PHYS 302: Unit 9 Quantum Mechanics Supplement – Athabasca University Unit 9 Bounded Waves: Quantum Energy Levels
After completing this Unit 9 supplement, you should be able to
1. identify that confining a particle leads to discrete energy levels for the particle and quantum states similar
to normal modes. 2. determine the allowable wavelengths and energies for a particle in a box. In the previous unit we saw that the wavefunction ψ(x,t)=Asin(kx-ωt) describes a wave of infinite extent if it is in
free space and able to move anywhere, making this equation valid everywhere. This is somewhat problematic if a
particle is localized instead. We can consider a particle “in a box,” which cannot leave (due to high walls). This is
analogous to a ball in a well, where in classical mechanics, if the total energy at the bottom is lower than the
potential energy at the top, the ball is trapped. In quantum mechanics, the square of the wavefunction represents the
probability, so the wavefunction is zero in regions where the particle cannot go (i.e., there is zero probability of
finding it where it cannot go). Strictly speaking, this requires that the potential energy outside the “well” be infinite,
since in quantum mechanics a particle can penetrate walls if they are not infinite, whereas a particle in classical
mechanics could not. We will say no more about this aspect of quantum mechanical motion. Simply consider our
problem to be that of a particle in a box with infinitely high walls. Since the probability is zero outside the box, and
we expect (require) the wavefunction to be continuous, the wavefunction must be zero at the walls. Thus, we have
the same situation as for a string with fixed ends in one dimension, a membrane fixed at the edges in two
dimensions, or a box with zero boundary conditions in three dimensions. We will consider only the one-dimensional
case here. Using normal mode solutions, if one fixed end is used as the origin, the solutions are in sinkxcosωt. If the
length of the box is L, then we require kL=nπ to meet the boundary condition that the wavefunction is 0 at x=L, that
is, k n
L . Then, p k n
L , and the kinetic energy of the states is K p2
2m 2 n 2 2
2 mL2 . Inside the box, there is no potential energy contribution, so apart from the rest mass, this is the energy of the allowed states. It is possible
for a particle to exist stably only in the discrete states, much as a bounded string, a membrane, or a resonating box
has a preferred state or normal mode. We also found that, in general, the discrete states have a characteristic energy.
Particles can change between states through the emission or absorption of radiation. Between any pair of states, a
specific amount of energy (and only that amount of energy) is associated with transitions. If the energy differences
correspond to the energy of light waves, then a characteristic spectrum, determined by the dimensions of the “box”
would be emitted or absorbed.
We have shown that particle waves allow a confined quantum system to have discrete energy levels, a fact which is
known for atoms. However, our assumption of bounding by an infinite potential does not allow for a very good
analogy to actual atoms. The force binding atoms, due to the positive charge in the nuclei of atoms, is quite close to
an inverse square law like that for gravity. In this case, the potential falls off proportional to the inverse first power
of distance from the nucleus. This does not resemble the potential of an infinitely deep well very closely. The
solution for even the simplest atom requires a mathematical treatment beyond what is presented in this course. The
particle-in-a-box model, with its spacing of energy levels, corresponds fairly well to atomic nuclei in some cases.
For nuclei, the energy levels are spaced by such large amounts that interactions with electromagnetic radiation are
usually in the gamma ray region of the spectrum. Nevertheless, energy levels for electrons in atoms also arise
(basically) because particle waves are confined in a finite region. Watch Video 4: Quantum Mechanics - Chapter 4 from the Cassiopeia Project. x mv means that a localized particle must have a large momentum.
The Heisengberg Uncertainty Principle
In the example above, the momentum (which leads to energy) results from confinement in a well. In the video, the
confinement in an atom and the differences in knowledge of the electron cloud lead to energy differences. The
various states have different wavefunction shapes. Transitions between states are possible, and as in the old spectral
theory, energy differences determine the wavelength of light emitted or absorbed.
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This note was uploaded on 02/18/2011 for the course PHYS 320 taught by Professor Martinconner during the Spring '10 term at Open Uni..
- Spring '10