Gasiorowicz stephen. quantum physics download

Since the delta function always appears multiplied by a smooth function under an in- tegral sign, we can give meaning to its derivatives. The delta function is an extremely useful tool, and the student will encounter it in every part of mathematical physics.

For simplicity we start with discrete events. Consider the toss of a six-faced die. This, of course, grows with N. When we do not have a discrete outcome, we must deal with densities. To be specific, consider a quantity that varies continuously, for example the height of a population of stu- dents. We can make this discrete by making a histogram Fig. Nobody will be exactly cm or cm tall, so we just group people, and somehow round things up at the edges.

We may want a finer detail, and list the heights in 1-cm intervals or 1-mm intervals, which will continue to make things discrete, but as the intervals become smaller, the histogram resembles more and more a continuous curve. Let us take some interval, dx, and treat it as infinitesimal. The proportion- ality to dx is obvious: twice as many people will fall into the interval 2dx as fall into dx.

Instead of calling these quantities averages, we call them expectation values, a terminology that goes back to the roots of probability theory in gambling theory. We are often interested in a quantity that gives a measure of how the heights, say, are distributed about the average. The deviations from the average must add up to zero. We can, however, cal- culate the average value of the square of the deviation, and this quantity will not be zero. We may illustrate it by going back to the population of students.

This is just an example of a general rule that the probability of two or more uncorrelated events is the product of the individual probabilities. Supplement 4-A The Wentzel-Kramers-Brillouin- Jeffreys Approximation This approximation method is particularly useful when one is dealing with slowly varying potentials.

Exactly what this means will become clear later. This cannot be, which means that the approximation 4A-7 must be poor there. The way of handling solu- tions near turning points is a little too technical to be presented here. Powell and B. Nuclei are very complicated objects, but under certain circumstances it is appropriate to view nucleons as independent particles occupy- ing levels in a potential well. Nowadays one uses the size of the charge distribution as measured by scattering electrons off nuclei to get nuclear radii.

It is not clear that the two should be expected to give exactly the same answer. Our considerations do give us an order of magnitude for it. The exponential part of the fit differs slightly from our derivation, but given the simplicity of our model, the agreement has to be rated as good.

Again, simple qualitative consider- ations explain the data. That is why all attempts to make thermonuclear reactors concen- trate on the burning of hydrogen actually heavy hydrogen—deuterium. For the same reason, neutrons are used in nuclear reactors to fission the heavy elements. Protons, at the low energies available, would not be able to get near enough to the nuclei to react with them. Supplement 4-C Periodic Potentials Metals generally have a crystalline structure; that is, the ions are arranged in a way that exhibits a spatial periodicity.

In our one-dimensional discussion of this topic, we will see that this periodicity has two effects on the motion of the free electrons in the metal. We digress briefly to discuss this requirement more formally. Here q must be real, because if q had an imaginary part, a succession of displacements by a would make the wave function larger and larger with each displacement in one or the other direction.

More gener- ally, the relation between q and k is more complicated. This means that an electron propagates without a change in flux. We may now ask: What is the effect of this overlap on the energies of the electrons? To answer this question, we consider first a classical analogy. We solve it by a trial solution. For large N there are many such frequencies, and they can be said to form a band.

If we think of electrons as undergoing harmonic oscil- lations about their central locations, we can translate the above into a statement that in the absence of neighbors, all electron energies are degenerate, and the interaction with neighboring atoms spreads the energy values. The spreading of the frequencies is the same effect as the spreading of the energy lev- els of the most loosely bound electrons.

For atoms far apart, with spacing larger than the exponential fall-off of the wave functions, all the energies are the same so that we have an N-fold degenerate single energy. Because the atoms are not so far apart, there is some coupling between nearest neighbors, and the energy levels spread. The regions of x for which the curve lines outside the strip are forbidden. The horizontal line represents the bounds on cos qa, and the regions of x, for which the curve lies outside the strip, are forbidden regions.

Thus there are allowed energy bands separated by regions that are forbidden. The exis- tence of energy gaps can be understood qualitatively. The energy levels corresponding to these standing waves are degenerate. Thus a metal may have an energy band partially filled. If an external field is applied, the electrons are accelerated, and if there are momentum states available to them, the electrons will occupy the momentum states under the influence of the electric field.

Insulators have completely filled bands, and an electric field cannot accelerate electrons, since there are no neighboring empty states. This corresponds to the breakdown of an insulator. The semiconductor is an insulator with a very narrow forbidden gap. The band structure is of great relevance in solid state physics. We shall learn in Chapter 13 that only two electrons are allowed per energy level. In case a the electrons fill all the energy levels below the edge of the energy gap.

The application of a weak elec- tric field will have no effect on the material. The electrons near the top of the filled band cannot be accelerated. There are no levels with higher energy available to them. Materials in which this occurs are insulators; that is, they do not carry currents when electric fields are applied. In case b the energy levels are only partly filled. In this case the application of an electric field accelerates the electrons at the top of the stack of levels. These electrons have empty energy levels to move into, and they would accelerate indefinitely in a perfect lattice, as stated in the previous section.

What keeps them from doing that is dissipation. The lattice is not perfect for two reasons: one is the presence of impurities, which destroys the perfect periodicity; the other is the effect of thermal agitation on the position of the ions forming the lattice, which has the same effect of destroying perfect periodicity.

Materials in which the energy levels below the gaps are only partially filled are conductors. The width of the gaps in the energy spectrum depends on the materials.

For some in- sulators the gaps are quite narrow. When this happens, then at finite temperatures T, there is a calculable probability that some of the electrons are excited to the bottom of the set of energy levels above the gap. These electrons can be accelerated as in a conductor, so that the application of an electric field will give rise to a current. The current is augmented by an- other effect: the energy levels that had been occupied by the electrons promoted to the higher energy band called the conduction band are now empty.

They provide vacancies into which electrons in the lower band called the valence band can be accelerated into, Conduction band Gap Narrow gap Valence band Holes a b c Figure 4C-2 Occupation of levels in the lowest two energy bands, separated by a gap.

Electrons cannot be accelerated into a nearby energy level. These electrons can conduct electricity. The electrons leave behind them holes that act as positively charged particles and also conduct electricity.

These vacancies, called holes, propagate in the direction opposite to that of the electrons and thus add to the electric current. This is the situation shown in Fig. The technology of making very thin layers of compounds of materials has improved in recent decades to such an extent that it is possible to create the analog of the infinite wells discussed in Chapter 3.

The outer one has a larger energy gap than the inner one, as shown in Fig. The midpoints of the gaps must coincide1 for equilibrium reasons. The result is that both electrons and holes in the interior semiconductor cannot move out of the region between the outer semi- conductors, because there are no energy levels that they can move to. Such confined re- gions may occur in one, two, or three dimensions. In the last case we deal with quantum dots. The study of the behavior of electrons in such confined regions is a very active field of research in the study of materials.

In summary, one-dimensional problems give us a very important glimpse into the physics of quantum systems in the real world of three dimensions. Bernstein, P. Fishbane, and S. Gasiorowicz Prentice Hall, Fishbane, S. Gasiorowicz and S. Thornton Prentice Hall, There are, of course, many textbooks on semiconductors, which discuss the many devices that use bandgap engineering in great quantitative detail.

See in particular L. Solymar and D. Supplement 5-A Uncertainty Relations In our discussion of wave packets in Chapter 2, we noted that there is a relationship be- tween the spread of a function and its Fourier transform. Our result depends entirely on the oper- ator properties of the observables A and B. Supplement 7-A Rotational Invariance In this supplement we show that the assumption of a central potential implies the conser- vation of angular momentum.

We make use of invariance under rotations. The kinetic en- ergy, which involves p2, is independent of the direction in which p points. The central potential V r is also invariant under rotations.

We show that this invariance implies the conservation of angular momentum. To identify the operators that commute with H, let us consider an infinitesimal rotation about the z-axis. This parallels the clas- sical result that central forces imply conservation of the angular momentum.

They must therefore each be constant. We call the constant m2, without specifying whether this quantity is real or com- plex. We will label the polynomial as Pl z. These polynomials are known as Legendre polynomials. Let us first write the solution of 7B as Pml z. Feynman and H. See Problem 10 in Chapter The description is arrived at in the following way.

We now note that in the presence of a potential flux is still conserved. Let us now consider the special case of a square well. The above argument shows us that we only need to consider the phase shift, since at large distances from the well the only deviation from free particle behavior is the phase shift.

The ratio can be related to the phase shift. The first term on the left-hand side is kr l Al 1, 3, 5,. The coef- ficient of the leading power, zl, can be easily obtained from eq. We shall use the simple abbreviation m1, m2 for Y 1 2 l1m1Y l2m2. We merely state the results. The symbol eijk is defined by the following properties: a It is antisymmetric under the interchange of any two of its indices.

Two consequences of this rule are i When any two indices are equal, the value of eijk is zero. We may write the equation in a very suggestive way by using the Levi-Civita symbol in two contexts.

First, the symbol may be used to give a matrix representation of the spin 1 angular momentum S. We need both equations to obtain separate equations for E and B. Supplement A Conservation of Total Momentum In our discussion of angular momentum in Chapter 8 we found that the assumption of in- variance of the Hamiltonian under rotations led to the appearance of a new constant of motion, the angular momentum.

In this supplement we show that the assumption of in- variance under spatial displacement leads to the existence of a constant of the motion, the momentum. The potential energy will change, unless it has the form V x1, x2, x3,. In quantum mechanics the same conclusion holds. We shall demonstrate it by using the invariance of the Hamiltonian under the transformation 13A The invariance im- plies that both HuE x1, x2,. Let us take a infinitesimal, so that terms of 0 a2 can be neglected.

This is a very deep consequence of what is really a statement about the nature of space. The statement that there is no origin—that is, that the laws of physics are invariant under displacement by a fixed distance—leads to a conservation law. In relativistic quan- tum mechanics there are no potentials of the form that we consider here; nevertheless the invariance principle, as stated earlier, still leads to a conserved total momentum.

For light atoms it is possible to solve such an equation on a computer, but such solutions are only meaningful to the ex- pert. We shall base our discussion of atomic structure on a different approach. The varia- tional principle discussed at the end of Chapter 14 had the virtue of maintaining the single-particle picture, while at the same time yielding single-particle functions that incor- porate the screening corrections.

A more general approach is that due to Hartree. The total can then be set equal to zero, since the constraints on the fi ri are now taken care of. The equation 14A is a rather complicated integral equation, but it is at least an equation in three dimensions we can replace the variable ri by r , and that makes nu- merical work much easier.

The trial wave function 14A-2 does not take into account the exclusion principle. To take the exclusion principle into account, we add to the Ansatz represented by 14A-2 the rule: Every electron must be in a differ- ent state, if the spin states are included in the labeling.

A more sophisticated way of doing this automatically is to replace 14A-2 by a trial wave function that is a Slater determi- nant [cf.

The resulting equations differ from 14A by the addition of an exchange term. The new Hartree-Fock equations have eigenvalues that turn out to differ by 10—20 percent from those obtained using Hartree equations supplemented by the con- dition arising from the exclusion principle. It is a little easier to talk about the physics of atomic structure in terms of the Hartree picture, so we will not discuss the Hartree-Fock equations.

We may expect, however, that for low Z at least, the splitting for different l values for a given n will be smaller than the splitting between different n-values, so that electrons placed in the orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f,. Screening effects will accentuate this: Whereas s orbitals do overlap the small r region significantly, and thus feel the full nuclear attraction, the p-, d-,. This effect is so strong that the energy of the 3d electrons is very close to that of the 4s electrons, so that the anticipated ordering is sometimes disturbed.

The same is true for the 4d and 5s electrons, the 4f and 6s elec- trons, and so on. The dominance of the l-dependence over the n-dependence becomes more important as we go to larger Z-values, as we shall see in our discussion of the peri- odic table. Any vibration with a restoring force equal to hooke's law is generally caused by a simple harmonic oscillator. Gasiorowicz, quantum physics. The potential for the harmonic ocillator is the natural solution every potential with small oscillations at the minimum.

The key point is that the transmission probability decays exponentially with barrier width The potential for the harmonic ocillator is the natural solution every potential with small oscillations at the minimum. Fitzgerald, honda elsinore owners workshop manual: "quantum physics" third edition by stephen gasiorowicz wiley Almost all potentials in nature have small oscillations at the … Modern physics stephen g, let's be friends grade 1.

The key point is that the transmission probability decays exponentially with barrier width. We have shown that X is hermitian. There is actually a neater way to do this. However, H is hermitian, so that the eigenvalues are real. If H is not hermitian, then all four eigenvalues are acceptable. An examination of the equation involving v2 leads to an identical equation, and we associate the — sign with the b2 eigenvalue. We calculate as above, but we can equally well use Eq.

We now take Eq. We then use Eq. We use the fact that Eq. This follows immediately from problems 5 and 6. The quadratic terms change the values of the eigenvalue integer by 2, so that they do not appear in the desired expressions. This is known as the Poisson distribution.

In this calculation it is only the commutator [p 0 , x 0 ] that plays a role. We follow the procedure outlined in the hint. We can now integrate w. We can use the procedure of problem 17, but a simpler way is to take the hermitian conjugate of the result. Thus the two results agree. Consider the H given.

We again expect a cyclical pattern. However, we may well 2 2 2 2 have chosen the n direction as our selected z direction, and the eigenvalues for this are again 2,1,0,-1, By the same argument we can immediately state that the eigenvalues are 7m i.

The problem breaks up into three separate, here identical systems. The ground state energy correspons to the n values all zero. The procedure here is exactly the same. These are the only possibilities, so that we have three eigenvalues all equal to zero. Now the sum of the eigenvalues is the trqice of M which is N see problem Thus there is one eigenvalue N and N —1 eigenvalues 0. The set M1, M2 and M3 give us another representation of angular momentum matrices.

Now let U be a unitary matrix that diagonalizes A. The problem is not really solved, till we learn how to deal with the situation when the eigenvalues of A in problem 13 are not all different. The construction is quite simple. It is. We may use the material in Eq.

If the matrix M is to be hermitian, we must require that A and all the components of B be real. What may be relevant for a potential energy is an average, assuming that the two particles have equal probability of being in any one of the three Sz states. We need to calculate 1 1 2 2 the scalar product of this with the three triplet wave functions of the two-electgron system.

It is easier to calculate the probability that the state is found in a singlet state, and then subtract that from unity. The eigenfunction of the rotator are the spherical harmonics. This result should have been anticipated. The eigenstates of L2 are also eigenstates of parity. Since the perturbation represents the interaction with an electric field, our result states that a symmetric rotator does not have a permanent electric dipole moment.

The second order shift is more complicated. This can easily be seen from the table of spherical harmonics. The orthogonality of the spherical harmonics for different values of L takes care of the matter. The problem therefore separates into three different matrices. The kinetic energy does not change since p2 is unchanged under rotations.

The energy is the sum of the two energies. The final state is 5-fold degenerate, and the same splitting occurs, with the same intervals. What will be the effect of a constant electric field parallel to B? The perturbation acts as in the Stark effect. The effect of H1 is to mix up levels that are degenerate, corresponding to a given ml value with different values of l. There will be a further breakdown of degeneracy. The second order shift for the upper state involves summing over intermediate states that differ from the initial state.

A picture of the levels and their spin-orbit splitting is given below. The latter can be rearranged into states characterized by J2, L2 and Jz. These energies are split by relativistic effects and spin-orbit coupling, as given in Eq.

We ignore reduced mass effects other than in the original Coulomb energies. Under these circumstances one could neglect these and use Eq. The unperturbed Hamiltonian is given by Eq. These will be exactly like the spin triplet and spin singlet eigenstates. According to Eq. This is not surprising. The ionization potential for sodium is 5. Thus the possible states are S1 , P1, D1.

Thus the S and D states have positive parity and the P state has opposite parity. Given parity conservation, the only possible 3 admixture can be the D1 state. B term, if L is not zero. For the S1 stgate, the last term does not contribute. If the two electrons are in the same spin state, then the spatial wave function must be antisymmetric. The problem is one of two electrons interacting with each other. The form of the interaction is a square well potential.