The Physics of Energy, page 61
Box 15.2 The Semiclassical Approximation
In this box we derive the semiclassical approximation, which underlies the estimations of tunneling and barrier penetration found in eqs. (15.18) and (15.19). This analysis is included for the interested reader for completeness; a simpler intuitive explanation is given in the text.
We parameterize the solution to the Schrödinger equation in the form . Substituting into the time-independent Schrödinger equation (7.7), we find
where, between the first and second line, we cancelled the common factor of .
We can ignore higher-order “quantum effects” by dropping the term proportional to . Solving for we obtain
and
This approximation is valid when the magnitude of the dropped term is small compared to the term,
The semiclassical approximation can be applied not only in a classically forbidden region but also in any region where the potential varies slowly. In a classically allowed region, we can define a variable DeBroglie wavelength , where . The condition for the validity of the semiclassical approximation can then be expressed as . Thus, the approximation is valid if the DeBroglie wavelength is a slowly changing function of position; this condition is intuitively sensible in regions where the particle’s energy is substantially larger than the potential, but is also mathematically correct in classically forbidden regions, where the variable DeBroglie wavelength becomes formally imaginary.
Barrier Penetration Factor
When a quantum particle of energy E encounters a constant potential barrier of height and thickness d, it has a probability of tunneling through the barrier to the other side
In the semiclassical approximation, this result can be generalized to a barrier of variable height ,
Here, are the classical turning points, where .
The semiclassical approximation for tunneling through a classically forbidden region with a varying potential thus gives a barrier penetration factor
(15.19)
in analogy to eq. (15.17).
The semiclassical approximation can fail either when the potential is changing too rapidly or when and the (imaginary) De Broglie wavelength (Box 15.2) diverges. vanishes at the classical turning points, so the approximation is suspect near the limits of the integral in eq. (15.19). Fortunately, the integrand vanishes at those limits, so the region where the approximation fails makes a negligible contribution to the barrier penetration factor. Figure 15.4 sketches regions where the semiclassical approximation is or is not valid for a complicated potential. Examples of barriers and barrier penetration factors are given in Box 15.2.
Figure 15.4 A fairly complicated potential V(x). For the energy shown by the line labeled E, the regions where the semiclassical approximation holds are shaded in green. In the red shaded regions – either near classical turning points at and or in regions where is not small – the semiclassical approximation breaks down.
15.4Tunneling Lifetimes
The barrier penetration factor estimates the probability that a quantum particle encountering a barrier will pass through it. Usually we want to compute something slightly different, namely the probability per unit time that a particle initially in a trap will tunnel out of it. Although a full quantum mechanical treatment of this is difficult, it can be estimated in the spirit of the semiclassical approximation simply by multiplying the probability of tunneling by a classical estimate of the frequency of encounters between the particle and the barrier.
Consider a particle trapped in the potential shown in Figure 15.5. Classically, ignoring tunneling, it simply oscillates back and forth in its trap. We can calculate the period of this oscillation. Given the particle’s energy E, the speed of the particle inside the potential is given by
(15.20)
The period of the classical motion is the time it takes the particle to make one full cycle of its motion, or twice the time it takes the particle to go from one turning point to the other. This can be computed by integrating between the two turning points and multiplying by 2,
(15.21)
Figure 15.5 A particle with energy E is trapped in a one-dimensional potential well. According to Newton’s (classical) laws it can only bounce back and forth between and . In quantum mechanics, however, it has a small but nonzero probability of tunneling through the barrier and being found to the right of .
The oscillating particle has a chance to penetrate the barrier once in each time interval T. So we estimate the barrier penetration probability per unit time, , to be
(15.22)
where comes from eq. (15.19) and comes from eq. (15.21). Often we are interested in a situation where barrier penetration results in the decay of a system such as a nucleus. In that case, the decay lifetime is given by the inverse of the barrier penetration probability per unit time. Thus, in the semiclassical approximation, the decay lifetime is simply the period of the classical motion divided by the barrier penetration factor,
(15.23)
Example 15.2 Barrier Penetration
Tunneling through a barrier of constant height
This is the simplest case. If a particle of energy impinges on the rectangular barrier shown in the figure, then the probability that it will pass through is . Note that the tunneling probability decreases dramatically with the mass of the particle as well as the height and thickness of the barrier. Consider, for example, energy and distance scales typical of atomic physics. Imagine an electron of energy 2.5 eV incident on a 5 eV barrier of thickness 5 Å( m).
Tunneling of electrons through barriers with heights and widths typical of atomic energy scales is thus not so unlikely.
In contrast, under the same conditions, a proton with mass roughly 2000 times the mass of an electron has a probability of tunneling! This has practical consequences: when two pieces of metal are brought very close to one another, electrons can tunnel from one to the other (this is the physical basis of the scanning tunneling microscope or STM), but the atoms themselves, which are even heavier than protons, stay put.
Tunneling through a parabolic barrier
The potential is an approximation to any smooth potential hump near its maximum. Suppose a particle with energy comes in from the left. Classical mechanics would require that it bounces back, never getting further than its classical turning point at , where , or . Quantum mechanics allows the particle to tunnel from through to the other classical turning point at . From the barrier penetration formula, we estimate a tunneling probability,
where we have defined . (Even though the potential well is upside down, defines a natural energy scale for the quantum mechanics of a particle in the presence of this potential.)
Tunneling Lifetimes
The lifetime of a state that decays by tunneling can be estimated by
The barrier penetration factor can be extremely small due to the exponential suppression of tunneling, leading to lifetimes far longer than the period of the classical motion .
15.5The Pauli Exclusion Principle
The richness of atomic physics is due in no small measure to the fact that no two electrons can occupy the same state. This exclusion principle forces electrons in atoms into ever-higher energy levels with more complex spatial structure. The sequential filling of shells of available states is responsible for the prominent patterns in the periodic table of the elements. It is also responsible for a similar richness in the properties of nuclei and for the electronic properties of metals and semiconductors, which are exploited in photovoltaic devices §25.
The exclusion principle was discovered empirically by Wolfgang Pauli in 1925 as he attempted to explain the patterns of the periodic table using the (then novel) ideas of quantum mechanics. In fact, however, the exclusion principle itself cannot be derived from nonrelativistic quantum mechanics. The axioms of §7 do not specify the nature of the particles to which they apply. In principle, as far as quantum mechanics is concerned, every electron could be different with, for example, a different mass and/or electric charge.
Example 15.2 An Approximately Parabolic Trap and Barrier
Consider a particle of mass m trapped in a sinusoidal potential . We estimate the time it takes for the particle to tunnel from one minimum to the next. We approximate both the trap and the barrier quadratically.a Near each minimum the potential looks like a harmonic oscillator potential of the form , and angular frequency . We assume that the particle begins in the ground state of this oscillator with . The period of its classical motion is . We estimate the barrier penetration factor by approximating the barrier as an inverted parabola (as in Box 15.2), with and ,
So the tunneling lifetime is
The tunneling lifetime grows exponentially with the ratio of the height of the barrier to the ground state energy . Thus, for example, if eV, the trapped particle’s classical period is s, but if the trap is 10 eV high, the tunneling lifetime is approximately s.
aNote that this cannot be a good approximation for both the trap and the barrier, since the second derivative of the potential vanishes halfway up. This approximation is good enough, however, for an order-of-magnitude estimate of the tunneling lifetime.
When physicists in the 1930s began to combine quantum mechanics with Einstein’s special theory of relativity, they found that it is necessary to view all particles as quantum excitations of fields that permeate space. We have already introduced the classical electromagnetic field and alluded in §7.3.3 to the fact that photons are its quantum excitations. We return to this subject in more depth in §22 when we derive the spectrum of blackbody radiation. Similarly, electrons are excitations of the electron field, and their charge and mass are properties of the field. The fact that all electrons must be treated quantum mechanically as identical particles is one of the most powerful implications of relativistic quantum field theory, a subject that goes beyond the scope of this book.
The Pauli Exclusion Principle
When relativity and quantum mechanics are combined in quantum field theory, it emerges that particles come in two types: bosons, with integer spin, whose quantum wavefunctions are symmetric under exchange of particle coordinates; and fermions, with half-integer spin, whose wavefunctions are antisymmetric under particle exchange.
Fermions obey the Pauli exclusion principle, which requires that only one particle can occupy a given quantum state.
Beyond the identical quantum nature of the excitations of a general quantum field, there is a deep connection between the intrinsic spin of a field’s quanta and the symmetry of the quantum wavefunctions for these quanta under an exchange of particle labels. Particles with integer spin (the spin one photon, for example) can only exist in states symmetric under particle exchange, and particles with half-integer spin (electrons, protons, and neutrons, for example) can only form states that are antisymmetric under particle exchange. Such particles are known as bosons and fermions respectively. All the fundamental particles from which matter is built are fermions.
The exclusion principle for fermions follows immediately from the antisymmetric nature of fermion wavefunctions. Consider two electrons labeled by coordinates and . Here τ includes not only the electron’s space coordinate x, but also its spin along a fixed axis, which (§7) can take on values . Suppose the two electrons occupy two states a and b with normalized wavefunctions . The two-particle system is described by a two-particle wavefunction, , which must be antisymmetric under exchanging with , namely
(15.24)
It is easy to see that this wavefunction changes sign when and are interchanged. It is also manifest that the wavefunction vanishes if both particles are in the same state, i.e. if . Thus, fermions obey the exclusion principle, while any number of bosons can occupy the same state.
Often, to first approximation, the energy of a quantum state for an electron is independent of the particle’s spin. Such is the case, for example, for the Coulomb potential summarized in §7.8.2. Each energy basis state can then accommodate two electrons, one with spin and the other with spin . The orbital of eq. (7.51) is full when it contains two electrons, explaining the stability of monatomic helium. The next set of states, with , includes the orbital and the three orbitals. These four orbitals are filled with eight electrons, leading to another highly stable element, neon, with a total of ten electrons. Thus, the sequential filling of atomic orbitals builds up the periodic table of elements. In the following chapter (§17), we apply similar principles to explain some of the stability properties of nuclei.
Discussion/Investigation Questions
15.1 Classical systems also decay. The decay or failure rate of many systems follows a bathtub curve. Look up and consider this type of decay and contrast it with the decay pattern of a quantum system.
15.2 The scanning tunneling microscope (STM) makes use of quantum tunneling to obtain resolutions down to the scale of individual atoms. Investigate the operating principles and applications of STMs.
15.3 It is not hard to balance a (brand new) ordinary pencil vertically upon its eraser. This is a configuration of stable classical equilibrium. Explain why in principle, even under the best of conditions (flat surface, symmetric pencil, no wind currents), the pencil is unstable due to quantum mechanical tunneling. Give a very rough estimate of the tunneling time, and compare to the age of the universe.
Problems
15.1 The first radioactive element discovered by Polish physicist Marie Curie was radium-226. It has a half-life of 1700 years. Suppose you obtained a microgram of pure . How long would you have to wait before the number of decays per second from this sample fell to ?
15.2 Uranium-238 and uranium-235 both decay by α-particle emission. Their measured half-lives are years and years respectively. At the present time the ratio of to in naturally occurring uranium is about 0.72%. It is believed that the uranium on Earth was formed in a supernova that took place billions of years ago. It is also believed that the ratio of / produced in supernovae is roughly 1.65. Estimate how long ago the supernova that produced the uranium on Earth took place.
15.3 Potassium-40 () is a naturally occurring form of potassium whose β-decays are responsible for some of Earth’s geothermal energy (see §14.3 and §32.1.3). can decay in two different ways. The probabilities per unit time for the two decay modes are y and y. What is the lifetime of ? What is the ratio of the average number of decays of type one and type two?
15.4 A radioactive source is emitting on average 40 particles per second. In any particular one second interval, what is the probability that it emits 40 particles? 30 particles? None? What is the probability that it emits less than 40 particles?
15.5 Consider the ground state and the first and second excited states of a simple harmonic oscillator as described in §7. Verify the statement made in the text that the probability of finding the particle outside the classically allowed region is 16% when the particle is in the ground state, and compute the corresponding probabilities for the first and second excited states.
15.6 [T] Verify the barrier penetration factor obtained for the parabolic barrier in Box 15.2.
15.7 [T] A particle of mass m and energy E encounters a triangular barrier of width and height (see Figure 15.6). Estimate the probability that it will tunnel through the barrier. How does the probability compare with that for a rectangular barrier of the same height and width?
Figure 15.6 A triangular barrier.
15.8 [T,H] Compute the exact amplitude for transmission and reflection of a particle from a rectangular potential barrier of height and width d by matching the wavefunction and its derivative at the boundary points . Note that, for a barrier that is high and wide, the leading term has the form of eq. (15.16) with a multiplicative coefficient as anticipated in Footnote 1. Show that further corrections, which arise from the exponentially increasing solution in the classically forbidden region, are exponentially suppressed.
15.9 An experimenter seeks to use a sinusoidal potential of the form studied in Example 15.2 to make a one-dimensional trap to confine a number of electrons for further study. She places one electron in the ground state of each potential well. Each electron’s energy should be 0.1 eV and the electron’s tunneling lifetime should be at least five years. Using the approximations developed in Example 15.2, estimate the minimum necessary height of the potential (), the spacing d between minima in the potential, and the root mean square width of each electron’s probability distribution.
15.10 [T] The energy levels of the one-dimensional harmonic oscillator are given by with Suppose two spin-1/2 fermions are trapped in a one-dimensional oscillator potential. What are the energies and degeneracies of the first three energy states of this system? (Remember that the degeneracy is the number of independent states with the same energy.)
15.11 [T,H] The energy levels of the three-dimensional harmonic oscillator are given by with . Suppose three spin-1/2 fermions are trapped in a three-dimensional oscillator. What is the energy of the ground state? What is its degeneracy? (Remember that the degeneracy is the number of independent states with the same energy.) Extra challenge: What is the energy of the first excited state? What is its degeneracy?
