Quantum Theory of Tunneling

The following sections are included:

• Dedication
• Preface to the Second Edition
• Preface to the First Edition
• Contents
• Introduction

Three years after the discovery of natural radioactivity in 1896, Elster and Geitel [1],[3] found the exponential decay rate of radioactive substances experimentally. In 1900 Rutherford [2] introduced the idea of half-life of these chemicals, i.e. the time that the number of radioactive nuclei reach one-half of their original number. In 1905 Schweidler [4] showed the statistical nature of the decay. This means that the probability of disintegration of a nucleus does not depend on the time of its formation and also the time that a particular nucleus decays can only be predicted statistically. This idea was verified empirically by Kohlrausch [5] in 1906. Later experiments showed that the decay width Γ, which is related to the half-life τ by (or if Γ measured as energy), does not depend on external variables such as pressure, temperature or chemical environment…

We start this chapter with a description of the decay of a quantal system by tunneling which is an essential feature of a large number of problems. In such a decaying system the measurable quantity is the survival probability, i.e. the probability of finding the system at the time t in the same state ψ in which it was with certainty at the initial time t = 0…

In this chapter we study a number of analytically solvable problems, all onedimensional or reducible to one-dimensional Schrödinger equation. These solvable problems can be grouped as follows:

• (i) – Confining and nonconfining potentials with central barriers.
• (ii) – One-dimensional problems when the motion of the particle is not restricted along the direction of motion.
• (iii) – Those situations where a particle is initially trapped behind a barrier and in the course of time escapes to infinity…

In the previous chapter we discussed a number of exactly solvable tunneling problems for one-dimensional tunneling and in each case we found the wave function from the solution of the time-independent Schrödinger equation. Now we want to consider few time-dependent problems for which we can find the wave function analytically, assuming that the initial wave function, ψ(x, 0), is known. The form of ψ(x, 0) can be given in one of the following forms:

• (a) – The initial condition, ψ(x, 0), may be given as a wave packet located far to the left of the barrier, (the barrier is nonzero around x = 0) and after tunneling this wave packet moves to the far right.
• (b) – The initial condition can be given as a wave train, again to the left of the barrier, and a quantum shutter opening at t = 0 allows this wave train to tunnel through the barrier.
• (c) – For the general case of tunneling through a potential of finite extent when the transmission amplitude T(k) for the incoming plane wave e^ikx is known, we can determine the asymptotic form of an incoming Gaussian wave packet when this wave packet has tunneled and moved to the far right of the barrier.

The well-known semiclassical approximation called WKB method (after Wentzel, Kramers and Brillouin), is the most widely used approximation for solving tunneling problems [1]–[6], and while it is often applied to one-dimensional cases [7]–[10], it is possible to modify it in different ways to solve two- or three- dimensional tunneling. In this chapter we discuss this approximate technique in some detail and find conditions for its validity [11]. We will also consider another semiclassical approximation, that of the Miller-Good [12]…

In addition to the semiclassical methods that we studied in the previous chapter, there is the well known technique of the Bohr and Sommerfeld which can be applied to solve the problems of one-dimensional quantum tunneling. In this chapter we first discuss the application of this method to investigate the escape of a particle trapped behind a barrier and the approximate determination of the decay width Γ [1]–[4]. Then we show how in the case of a double-well potential, the approximate energies of the low-lying states can be calculated with the help of this method…

The theory of the α-decay of a nucleus by tunneling through the Coulomb potential was developed by Gamow [1],[2] who solved the Schrödinger equation approximately with outgoing wave boundary condition and found complex discrete eigenvalues [3]…

In the preceding chapter we studied some solvable examples where the wave functions were found analytically. Now we want to study the energy splitting and the motion of a wave packet in a double-well confining potential, and in particular to see the difference between the tunneling in a symmetric and an asymmetric potential. The level splitting caused by tunneling is an important feature of these potentials and plays an essential role in a large number of problems of molecular physics and chemical physics [1]–[5]…

At the beginning of Chapter 2 we defined tunneling as a quantal process which is forbidden by the laws of classical dynamics. Nonetheless we can describe the tunneling phenomena either partially, or completely in terms of the laws of classical dynamics provided that we replace the simple quantal system by a complicated interacting system. After solving and simplifying the result, we find a set of classical laws of motion derivable from a classical Hamiltonian function, however the interaction is not simple - it no longer looks like a potential barrier, and it also depends on the Planck's constant ħ [1]–[5]…

In condensed matter physics, the technology of molecular beam epitaxy and layer by layer growth of semiconductor heterostructures have given rise to numerous tunneling applications. In these structures the time-dependent barriers appear in two ways: An explicit time-dependence arises due to the applied a.c. fields with frequencies reaching into microwaves (10⁷ – 10¹¹ Hz). Secondly, resonant tunneling occurs in a small energy range and this can be modeled by considering an oscillating barrier [1]–[13]. In these problems mostly one-dimensional tunneling is important and therefore we study the motion of a particle of mass m in a barrier V (x, t), where we assume that

The close connection between the time-dependent quantum scattering theory and the decay of a quasi-stationary state by tunneling is the topic that we want to discuss in this chapter…

We have already seen that in the case of time-dependent potentials the wave equation can be written as an infinite set of coupled differential equations, Eq.(10.15). Similar set of equations results from the problem of tunneling of a composite system, e.g. tunneling of a bound molecule in one dimension (Chapter 21). Even in the simple case of one-dimensional tunneling from a nonsymmetic potential, we have a coupled channel problem [1]…

In the preceding chapters we have studied certain aspects of the one- and three- dimensional tunneling using the Schrödinger equation as the starting point of our investigation. In this and the next two chapters we examine other methods of formulating and solving tunneling problems…

In addition to the formulations of quantum tunneling in terms of the wave equation and by the path integral method, a third method, that of the Heisenberg equations can also be used to investigate the motion of the particle under a barrier. The similarity between the Heisenberg and the classical equations of motion in some cases helps us to a better understanding of the problem, specially when the Hamiltonian (or Lagrangian) of the corresponding classical system is difficult to find or is not unique [1],[2]. At the same time we have an alternative method of solving tunneling problem which is particularly useful in the formulation and the solution of the dissipative quantum tunneling problem [1]. Also a semiclassical method of approximation based on the Heisenberg equations can be formulated in which these equations are combined with the classical equations of motion [3]. This method has been applied to the quantum tunneling in cubic and quartic potentials [3],[4]…

We have already seen that a classical description of tunneling is possible at the expense of introducing an artificial system with infinite degrees of freedom (Chapter 9). Another approach which is also helpful in establishing a connection between quantum mechanical and semiclassical formulations of tunneling is based on the distribution function…

When we were discussing Gamow's theory and the radiation boundary condition in Sec. 7.1, we pointed out that the radiation boundary condition shows exponential growth as a function of the radial distance and therefore is not a square integrable wave function. This is true whether as in Gamow's theory of α-decay we use semiclassical approximation or solve the problem exactly. In addition in Chapter 11 we studied the connection between the properties of the scattering matrix and the tunneling problem. For the scattering of a particle from a center of force, the S-matrix has simple poles in the complex E-plane and the position of these poles on the negative real E-axis are the bound state eigenvalues. For the continuum spectrum, the S-matrix has a cut along the Re E > 0 [1]–[3]…

In our introduction to the subject of quantum tunneling we mentioned that when the energy of the particle is less than the maximum height of the barrier, the motion becomes forbidden classically, but such a motion is possible in quantum mechanics. For one-dimensional tunneling the presence of insurmountable barrier is always the case for tunneling. However in two or three dimensions we can have a situation called “dynamical tunneling” where the energy of the moving particle makes a part of space inaccessible classically, but tunneling through this region is possible [1],[2]. A solvable example illustrating this type of phenomenon will be presented in Sec. 25.7…

In this and the next two chapters we will review various attempts to answer the question: “How long does it take a particle or a wave packet to move from one side of a barrier and appear on the other side if the energy of the particle is less than the maximum height of the potential barrier?” Recent progress in fabrication of semiconductor structures in the nanometer range has brought this problem to the forefront of research. But as we will see, no satisfactory answer has so far been found…

One of the controversial issues of modern quantum theory is the question of tunneling time, i.e. the time that takes a particle to move from one side of the barrier to the other side. We can classify the problems related to the tunneling time into two groups:

• (1) – In the first group we have a well-defined formulation and an unambiguous way of measuring the time that is related to the motion under the barrier. among these are the following examples:
  • (a) – Oscillations of a wave packet in a double-well potential such as the period of motion in a NH₃ molecule, Sec. 25.5.
  • (b) – Lifetime of a particle or system of particles trapped behind a barrier, and by tunneling can escape to infinity, e.g. α-decay.
  • (c) – A lesser known, but the case of reection time which allows for an exact quantum-mechanical formulation, and will be discussed later in this chapter.

We have already seen that a mathematically acceptable formulation of quantum mechanical reflection time is possible and such a formulation was given by Bracher and Kleber [1]. This proper formulation was achieved by constructing a self-adjoint operator T_R, (19.151), and this operator commutes with the Hamiltonian and measures the reection time. But for the problem of tunneling time no such a self-adjoint operator is known. We have also discussed the problems associated with the measurement of delay time in tunneling using the motion of the center or the peak of the wave packet as an indicator of the travel time…

The problem of tunneling of a system of particles with internal degrees of freedom has not received much attention, although in some important physical systems for example α-decay the decaying part is a composite system. In physics and chemistry of molecules there are observable effects attributable to the tunneling of a molecule from one well to the other in symmetric and asymmetric double-wells [1]–[7]. We have also seen (Sec. 20.5) an attempt to use the tunneling of a molecule as a way of introducing time operator and measuring tunneling time [8].

The passage of a quantum particle through a waveguide with variable cross section causes a shift in the phase of its wave function and the presence of a bottleneck in the waveguide acts like a potential barrier. For wave numbers corresponding to propagating modes there is reflection and transmission similar to the motion of a particle over the potential barrier. However for smaller wave numbers, below the cutoff frequency, the wave will be evanescent wave, i.e. it exhibits exponential decay which is one of the characteristics of quantum tunneling. Now let us consider the propagation of a scalar field, e.g. the electric (or magnetic) component of the electromagnetic wave in a similar waveguide. In both cases if the frequency of the incoming wave is above the cutoff frequency for a given model, the energy (or the probability current) can be transmitted with negligible attenuation. However if the frequency is below cutoff, that particular mode will be attenuated with reduced energy (or current). For the sake of simplicity here we will consider only TE_0,1 (the lowest transverse electric) of a microwave in which the electric field is orthogonal to the direction of propagation, our aim is to use this analogy and verify the validity of tunneling times experimentally [1]–[11]…

For most of the problems where quantum tunneling plays a major role, whether the physics of condensed matter or in nuclear physics, the velocity of the tunneling particle is very small compared to the velocity of light. However it is important to know the relative size of the relativistic corrections in any physical situation, and this relativistic correction is the subject that we will address in the first section of this chapter.

We have, up to this point, discussed extensively the problem of determination of the reflection or transmission amplitudes and coefficients for any given potential barrier as a function of the energy of incoming particle, or its angular momentum. Now we want to ask the following questions: Given an observable quantity such as the transmission coefficient as a function of energy or of angular momentum, what can we infer about the shape of the potential barrier? Also under what kind of conditions, this determination of the form of the barrier from empirical results will enable us to find a unique barrier? By considering tunneling as a special case of quantum scattering theory (Chapter 11) we can apply various techniques of the inverse scattering theory to construct the barrier given the transmission or reection amplitudes [1]…

There are numerous examples in the physics of atomic and molecular physics where quantum tunneling plays an important role in the explanation of the observed behavior of the system. We have already seen that the splitting of the energy levels in a double-well potential such as in ammonia molecule is due to tunneling [1]. We will discuss the energy levels of this molecule later in this chapter. We also study in greater detail some of the other simple problems in atomic and molecular physics where tunneling dominates the dynamics of the system.

Among the very large number of tunneling problems in solid state physics [1]–[3] we will briey discuss four very important and yet conceptually simple problems. The first one is about the motion of electrons in a periodic potential which is similar to the problem of tunneling through a series of identical barriers discussed in Sec. 3.3, only now the number of barriers is infinite. Then we study the tunneling of electrons in a simple metal-insulator-metal structure. While, in principle, this should be formulated as a many-electron tunneling problem and the current associated with it, we observe that one encounters certain difficulties with such a formulation and for this reason we replace it with a very simple model for such a tunneling. We then turn our attention to an elementary theory of electron tunneling through heterostructures. Finally we study the Josephson effect for both d.c. and a.c. currents.

As we mentioned in our historical review of the subject of quantum tunneling, in 1908 R. E. Rutherford showed that α particles are indeed the nucleus of He⁴. One of the great successes of quantum mechanics beyond the accurate prediction of the energy levels of hydrogen atom was the explanation of the α decay as a tunneling problem [1],[2]. Since the early days of the discovery of α decay of various radioactive materials, there has been a large body of literature about the theoretical prediction of lifetimes and modes of decay of these nuclei. For more recent theoretical works related to alpha decay see refs. [3]–[6]…

The following sections is included:

• Index