20 Quantum or Classical? A density of states perspective. (15 polnts) Another me
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20 Quantum or Classical? A density of states perspective. (15 polnts) Another metric that is often useful in determining if classical mechanics can be applied to a problem is the degree to which the granularity of the quantum states is relevant. A handy example comes from our familiarity with U.S. dollars, which are currently still quantized in units of of the penny $0.01. If one has only 10c, then the quantized nature of money is relevant, but if one has millions of dollars, then it is likely not so important. Often an estimate can be made using thermodynamic arguments. In thermal equilibrium, a system in contact with a "bath" at temperature T with have significant (incoherent) population of states within approximately kgT of the ground state of the system, where ka is Boltzmann's constant. When the number of states thermally populated is much larger than 1, then it is often reasonable to treat the system classically.2 Thus, a quick metric for determining if a classical or quantum treatment is appropriate is the ratio of the energy level spacing to the available thermal energy kBT. IfkpT » E then often a classical description is appropriate. For this problem, estimate (order of magnitude is good enough) the ratio kaT/AE for the systems below: IIn the intermediate regime between thee two limits, clever people have developed an array of so-called semiclassical methods, which are easier to use than the Schrödinger equation, but capture some of the essential quantum features of the motion. The WKB method in the Griffiths book is an example of this. 2There is, in fact, a way to deal with such systems, and departures from (boring) thermal equilibrium that can occur upon exciting such systems, using the quantum formalism of density matrices - but this is beyond the scope of this course 10 a) The moon confined to a ring of radius R, where R is the average distance between the center of the moon and the center of the earth. The surface temperature of the moon varies widely depending on illumination, but the core temperature is steady at around 1700 K. b) A hydrogen atom at room temperature. Use the Rydberg formula for the energy levels of the hydrogen atom and take E to be the energy difference between the ground state and the first excited state. c) An electron in aluminum experiencing thermal fuctuations around the Fermi energy (kinetic energy- 11.6 eV). Hint: Look up the expression for the density of states per unit energy in a metal, and multiply the density of states times kyT to get a rough estimate for the number of states populated. d) A 3.0 GeV electron, with 0.1% energy spread, confined to the NSLS-II storage ring with a circumference of 792 m. For this case use the beam's intrinsic energy spread instead of the thermal spread. Hint: What is the energy level spacing for particles confined to the ring at 3 GeV and how does this compare to spread of energies in the electron beam. e) A neutron confined to an 56Fe nucleus (z 7.5 fm) at room temperature. Hint: model the nucleus as a "bor" confining the neutron. What is the energy level spacing.Explanation / Answer
a)
The statement is quantum, because, the case of a particle in a one dimensional ring, that is, moon in a ring is equivalent to the particle in a box.
b)
The statement is again a quatum, because, quantum mechanics is a science of the very small. It explains the behaviour of matter and its interaction with energy of atoms. This statement is in accordance with the Bohr atomic model.
c)
This statemtn is a quantum mechanical statment, because, in quantum statics a branch of physics, Fermi dirac describes the distribution of the particles over energy states in systems consosting of identical particles that obey Pauli exclusion principle.
d)
It is a classical process where the heavy metal target is bombareded with pulses of highly energetic protons.
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