PHYSICS 430 NUCLEAR PHYSICS
READ: Krane,Chapter 2: 1, 2, 3, 4, 5, 6
This chapter should provide us with a sufficient introduction and/or review of the basic foundations of non-relativistic quantum mechanics for most of the topics in this course. The formalism is cast in the context of the Schrödinger equation. That this is applicable to most of the particles in the study of low energy nuclear physics is clear when one considers the energies involved and the masses of the particles involved. By contrast, however, we need to remember that the electron has a very small mass compared to its kinetic energy in many nuclear processes in which it is involved (e.g., radioactivity) and consequently needs to be treated as a relativistic particle. Moreover, the photon and neutrino have zero rest mass, and are therefore completely relativistic ( = v/c = 1). So, it behooves us to consider the physical situation to which we propose to apply the non-relativistic quantum mechanical formalism outlined in this chapter. The following comments are intended to guide and prompt your reading.
2.1We are reminded here of the pioneering work of Planck (blackbody radiation - emission) and Einstein (photoelectric effect - absorption) whose respective interpretations of the observations demonstrated the quantization of the EM radiation into energy quanta of individual energy E = h. This led to the wave-particle interpretation for light in genuine theoretical terms. Moreover, by special relativity, E2 = p2c2 + m2c4. For light, the rest mass m = 0.0 and therefore, the momentum of a light quantum (photon) is simply p = E/c. Applying this to the Planck relation, we find that for light, p = h/c, or p = h/. (These relationships should be familiar to you; we will use them repeatedly during the course.)
Louis deBroglie extended this interpretation (insightful symmetry!) by asserting that "particles" should also exhibit "wave-like" behavior such that their observable wavelength is = h/p. He turned out to be correct as initially verified by Davisson and Germer. (I suggest that you get a
3"x 5" card - or find a place in your notebook or textbook where you can write relationships you should commit to memory. Those above should be among them.)
This may also be the time to remind you of the classical concept of a "wave number" k, which is the number of wavelengths in 2, i.e., k = 2/. Using this, we can see that p = k, where
= h/2. Be reminded that the units of k are (length)-1.
The remainder of this section is dedicated to the profoundly important Heisenberg principle which applies to inherent limitations in the precision with which we can possibly make certain simultaneous measurements. The exact form of this relationship, i.e., that the product of the uncertainties is /2 assumes a gaussian form for the distribution functions (later to be synonymous with the "wave function") describing the object being measured. Two comments are apropos. First, the numerical size of h sets the physical scale for not only the uncertainty relationship, but the scale on which the quantum effects become important. Second, we shall refer often to the Heisenberg relationships as limiting conditions in qualitative and quantitative interpretations of physical situations. (I shall refrain from a long list here; you are forewarned!)
2.2This is the Schrödinger equation and its interpretations. If some of this appears terse, that is in part the summary nature of what is being done. But, it is also because the Schrödinger equation is and the interpretations one attaches to the state function, etc., are not obtained (derived) from some fundamental theoretical formalism; they are axiomatic. Two comments here. First, it is amazing that all this works so well when applied to appropriate physical systems - but it does! Second, this is not the only interpretation of the microworld, i.e., a completely non-deterministic, probabilistic nature. For example, the work of Bohm (c.f., Scientific American ...) has been shown to work and in some people's view, is preferable. No one has devised an experiment which clearly shows the validity of one as opposed to the other.
Herein are crucial definitions and concepts and you should do your best to internalize the significance of each.
•The distinction between the time-dependent and the time-independent SE.
•The time-independent equation is an eigenvalue equation; the eigenvalue is the total energy.
•The SE can be viewed as a statement of conservation of energy.
•The wave function (or the state function) tells you all you can know about the system.
•The potential function V(x) or V(r,t,...) contains all the physics of the system. That is, it is the potential function which distinguishes one physical system from another physical system.
•The state functions will exist in different parts of a physical system. Elementary conservation of probability demands the continuity conditions in (2.6).
•The interpretation of probability density for the state function is central. (2.7)
•Most state functions can be easily normalized (and should be to make quantitative sense). (2.9)
•Measurements of a physical quantity yield the expectation value. (2.10) You should see that this is little more than the linear average of a quantity -- except that in quantum mechanics, every observable physical quantity has an associated physical operator. When this operator acts on a state function, it has the property that it returns the corresponding physical value according to an eigenvalue equation. Hence, the operator is used as in (2.10).
•The continuity equation (remember from electricity and magnetism) characterizes a relationship between the probability density amplitude function and a probability density current j. (2.12) We will remind you of this more clearly in class.
•The SE is always a three dimensional equation. However, in some cases, if the potential function varies only along one dimension, the solution along the other dimensions is a constant. Which 3D representation one chooses depends on the symmetries of the physical system.
2.3Here are two classes of problems in one dimension:
(a) traveling waves (particles)
•free particle, i.e., no force on the particle no potential function over this region.
•particle encountering barriers, i.e., places in space where a force acts on the particle.
The effect of the force is to do work on the particle so as to change its kinetic and potential energy, keeping the total energy constant. The essential elements here are to recognize that the particle is represented in each "region" by the SE, but with a different potential function. Therefore, the state function (wave function, eigenfunction) in each region is different, but continuous across the boundary separating the regions (i.e., where the force acts). At any boundary encountered by a wave, the wave divides. Some is reflected backward; some is transmitted forward. Of particular interest is the relative amounts which are reflected and transmitted, i.e., the transmission coefficient (T) and the reflection coefficient (R). By conservation of probability (particles), R + T = 1.
Q: How does one normalize the free particle wavefunction? Q: Why is the conventional normalization not workable?
The concept of barrier penetration will be important when we consider alpha decay in detail. One finds that the alpha decay would not occur in naturally occurring radioactive nuclides were it not for this phenomenon.
For systems in which the potential function is a constant over some range in the motion, the SE always reduces to a simple form (2.24) and the solution to (2.24) can always be written as (2.25). The choice of the constant ki2 = [2m(E-Vi)/2] where i is the region number, is not altogether an accident; you should convince yourself that this k is the same as the wave number k above, i.e.,
k = p/ . Now, using the form (2.24) and the solution (2.25) for the case of a constant potential, you should see that you have two possible conditions:
(E-Vi) > 0 (i.e., ki2 > 0 pi2 > 0 > 0 , i.e., a classically physical kinetic energy), and,
(E-Vi) < 0 (i.e., ki2 < 0 pi2 < 0 < 0 , i.e., a classically impossible kinetic energy). In the latter case, if ki2 < 0, then
ki = i
Then, we have on substitution into (2.25),
= C exp(iki) = C exp[ii], and similarly for the second term whose amplitude is D. This then reduces to,
= C exp.
Note that this gives the correct solution (2.34) for the case (E-Vi) < 0, without having to remember to write the SE in the different form, (2.33). It is a matter of preference; I prefer this approach to understanding it.
Thus, if (E-Vi) > 0, (positive kinetic energy) the solution is an oscillating solution;
and, if (E-Vi) < 0, (negative kinetic energy) the solution is an exponential solution.
(b) confined waves (particles).
•square "well" potential function (abrupt vertical sides) whose depth is either infinite (first case) or finite (second case).
•smoothly varying sides (e.g., varying quadratically with x).
Q: After you have established the potential function, and then after you have found the general analytical solution to the SE for a particle confined to a , what conditions allow you to find the numerical values of the eigenvalues and remove all of the arbitrary constants in the solutions?
The case of the infinite square well is quite easy to follow and you should be able to work it through in detail (easily) to see exactly how you establish all the constants of the solution. The case of the finite square well is a bit more work; look this over more qualitatively to get the sense of the method. (You certainly could do it if you wanted to spend the time, but I do not think it is that important at this point.)
The case of the harmonic oscillator potential is certainly interesting, but the details are beyond the scope of this effort. Q: What is the physics motivation to use a potential of the form V(x) = kx2/2? One significant difference between this potential and most others is that the spacing of the energy levels (energy eigenvalues) is uniform. Note also that the lowest possible energy for any of these systems is not zero.
2.4The extension to three dimensions is really just that: an extension of the one dimensional formalism to three dimensions. There are four illustrations, given in various degrees of completeness.
Infinite Cartesian Well: Q: If you were to draw such a three dimensional object, what would it look like? I expect that you can work through this in sufficient detail (try it!) to show that if the potential is a constant (taken to be zero) the state function(s) which satisfy the SE (2.54) can be obtained by writing as the product of three functions, each depending only on (x), (y), and (z) respectively, i.e., separating the variables. Substitute and work this through as we did in class for the case of the one-dimensional case where we separated (x) and (t); the technique is the same. Note that there are three independent coordinates; therefore, there are three independent integers for which the boundary conditions in each dimension can be met. Obtain the total (product) state function with normalization (2.55) and obtain the total (sum) energy eigenvalues E. Q: What specifically (in clear words) is meant by degeneracy? Q: How many unique, but degenerate states occupy the third excited state of the system? (The ground state is the lowest state; the first excited state is the first one above that; etc.) Note that the state of the system, i.e., the state function, can be completely specified in terms of three integers and the dimension of the cube, a. So, you should be able to write the individual (unique, but degenerate) state functions for each state as (nx, ny, nz).
Spherical Well: The significance here is as follows... A straightforward, but not uncomplicated transformation will allow you (don't begin unless you have a chunck of time to give to it!) write the Laplacian operator 2 in spherical, polar coordinates, (r,). This would imply that the state function would also be a function of these variables, (r,). Now, the next part is very important! In general, the potential function will depend on all three spatial coordinates and on time. In keeping with our consideration of time-independent potentials, we would have V(r,). However, in many physical situations, the potential function will depend only on the radial distance from the center, V(r), and have no dependence on the angles (). This means that the force F = f(r) r/r, i.e., a "central force" whose size depends only on the distance from the center and whose direction is along the vector r.
So, if V depends on r only, then the state function (r,) can be written as a product solution as noted on page 27. That the variables can be separated is not obvious until you actually do it (which you will not do unless you have 2 in spherical, polar coordinates.) So, you take it on faith that the proper differential equations for () and () are equations (2.57) and (2.59) respectively, once and forever! The result for () is (2.58) which depends on one integer number to distinguish one azimuthal state function from another. The solution for () is obtained by solving (2.59). Note that the integers ml (for the part of the state function) and l (for the part of the state function) are coupled as noted below (2.59). The product wave function Y() = () (), properly normalized are known as the spherical harmonics, some values for which are noted as (2.60). The total state function (r,) = R(r) Y() is valid for any physical system in which the particle is moving in a central force field, i.e., with a central potential V(r). The state functions for different central potentials (i.e., different central force fields), will be entirely contained in the R(r) part of the state function. We have the functional dependence for the angle variables finished!
Associated with each degree of freedom there is always a quantum number. For example, in the Cartesian case, a point particle will have three degrees of freedom (x,y,z) and therefore three quantum numbers, one associated with each degree of freedom (nx, ny, nz). In the spherical case, the three degrees of freedom (r,) have associated quantum numbers (n, l, ml). In the spherical harmonic wave functions we will always have for each l value, a range of (2l + 1) ml values:
l ml l . The allowed values for l for a given nl value will depend on the form of the radial wave function, which in turn will depend on the functional form of V(r).
Note in equation (2.60) that there is another term which appears added to V(r). This appears here due to the separation constant between the angle part and the radial part of the state function. Thus, there is something added to V(r) at each value of r. Q: The units of 2 are consistent with units of angular momentum squared. (Check it!) The center of mass of the particle m has a rotational moment of inertia I = mr2 about the origin. So, if the numerator is L2 = I22 one gets for the added term 0.5 (I 2) which is the rotational kinetic energy of an object. Hence, this term is often called a "centrifugal" term, and it plays a role in radioactive decay, especially in alpha decay.
Now, there are three central potential illustrations: V(r) = 0.0 for some r < a, V(r) = for r > a (the infinite spherical well), V(r) = -kr2/2, the three dimensional harmonic oscillator, and V(r) = k/r, the coulomb potential. We will discuss the solutions for these three briefly in class. For any of these illustrations, the predicted probable locations for finding the particle as a function of r is given by 2.63 (although the representation is a bit strange on the first line.) Be sure that you qualitatively understand the significance of the different plots in Fig. 2.12 and Fig. 2.13 and that you can express these in words. For the cases where V(r) 0, the radial quantum number n often (but not always) places limits on l. For example, see the wave functions for the harmonic oscillator potential or the coulomb potential.
2.5What we would like to know about the particle is the three components of L, i.e., Lx, Ly, Lz.
Q: Why is it impossible to determine these components simultaneously? (In QM language, these operators do not commute, and therefore, there are not simultaneous stationary states of all three components.) Q: What two quantities do we use to describe the quantum mechanical state of the system? (L is the "orbital" angular momentum.) Q: Do you know the spectroscopic notation used to identify the angular momentum quantum numbers l? Q: What is the intrinsic angular momentum called? Particles with s = 1/2 are known as "fermions", or "fermi particles". These include the electron and the nucleons. Now, if one has both spin angular momentum s, and orbital angular momentum l, one must combine these to obtain the total angular momentum j. (We are here considering the single-particle states.) The technique is shown in equations (2.71 ff). You should be able to draw vectors to indicate the combining of the two angular momenta.
2.6The parity concept is a useful one in quantum mechanics since it can be shown that the parity operator yields a parity eigenvalue (or expectation value) which is not time dependent if the Hamiltonian is not time dependent. Therefore, the parity eigenvalue is a "good" quantum number for the system, i.e., it can be used to characterize the state of the system - permanently. Q: What is meant by a state function being either even parity or odd parity? Q: What is meant by a parity violation? Q: Which of the four fundamental forces has been shown to violate parity? Q: For a system described by a state function R(r)Y(), how can one determine the parity state of the system in trivial fashion?
01/29/2019 11:09 AM