PROBABILITY CURRENT DENSITY AND THE WKB APPROXIMATION

wavefunction of the form e^±ipx/l>. So it is not always possible to use the above product. There is however a formula that links directly each % with the corresponding /, without the need to invoke the concept of velocity or momentum.

Let us see how this is done. The time dependent Schroedinger equation reads in one dimension

Taking the complex conjugate

Now we focus our attention on a rectangular volume of space bound between the planes at points *1, x₂(see figure 1.14), and we ask about the variation with time of the total probability in that volume.We initially assume only an x-dependence of the wavefunction ^(x.t). The probability of finding an electron in this volume of cross section A is

Its variation with time is given by

FIGURE 1.14 Geometry for the calculation of the time-dependence of probability density.

Using the time dependent Schroedinger equations 1.55a, 1.55b, we transform the right hand side (=RHS) of the above equation as

Note the change of sign above from transferring the i from the denominator to the numerator. We therefore get by cancelling the V*P terms and incorporating the cross section A as extra integrations

We come now to a 3-dimensional generalization of 1.57. We have written 1.57 in such a way, so that this generalization is straightforward. It’s obvious that the 3-dimensional differential element dxdydz is equal to the volume element dV and the 1-dimensional differential operator will go to the grad operator V. Therefore, we get in 3-dimensions ox

We now use the well-known formula transforming a volume integral to a surface integral

where S is the surface enclosing the volume V. We finally get

What does equation 1.59 tell us? It tells us that the rate of change of probability density inside a volume V is equal to the flux of a quantity out of it. Therefore, that quantity must be the negative of the probability density current. Hence we obtain for the current that is due to a wavefu notion Ч⁷,

For stationary states of energy £, the wavefunction will be of the form ^x¥(r,t) = i ((r))e⁺'^E,lt1 and the exponentials will cancel out, so that we can write

If we apply this formula for example to the incoming to the barrier wave, we immediately get by elementary manipulations in 1-dimension

since ftk/m is the velocity of the incoming electrons.

The main application of the formula 1.60b is not to derive again the expression for the current density in a simple case as the above but to derive an expression for the transmission coefficient of a general barrier and an expression for the current in the channel of a nanotransistor in later chapters of the book. For the time being we derive the required expression for the transmission coefficient of a general barrier without the resource of 1.60b and with a method not as rigorous as that based on 1.60b but quicker and more physically transparent. We will revisit this problem more rigourously in a later chapter.

Consider then a 1-dimensional general barrier characterized by the potential function V(x as in figure 1.15, and an incoming electron of energy £ for all x. The barrier can be divided into thin rectangular slices as shown again in figure 1.15. Now the wave- function of the electron must look like e'^kx far to the left and far to the right of the barrier so the problem is in this respect similar to the problem of section 1.7. We can imagine that this electron wavefunction goes through the succession of rectangular barriers into which the general barrier can be divided and each time it is attenuated by the transmission coefficient of the i^lh rectangular barrier i.e. by

where V, is the local barrier height of the i^th slice and Ax, is its width. We have omitted the preexponential factor as this is usually very small compared to the exponential. We also assume that there is no reflection at each of the individual slices of the barrier.

FIGURE 1.15 A wave incident on an arbitrary barrier can be thought to be transmitted through the rectangular layers into which the barrier can be divided.

The total transmission coefficient T will be the product of all the T„ that is where N is the total number of slices into which the barrier has been divided. Therefore

But in the limit of very large N the summation is simply an integral. Hence

where x_b x₂ are the points that define the barrier, i.e. x_h x₂> are such that in the interval [x_ux₂ we have E Formula 1.64 can be obtained by a more polished version of 1.60, that is, by an approximation called the WKB approximation from the names of the authors of the original papers. We will prove this approximation later in the text and use it in both the theory of electron emission and the nanotransistor, but it will be sufficient for now to say that it is strictly valid for 1-dimensional problems and it is only valid for “deep tunnelling”, i.e. when the energy of the electron E is well below the barrier maximum.

PROBLEMS

• 1.1 Show that the nodes of the intensity in the two-slit experiment with infinitesimal width slits lie on a hyperbola.
• 1.2 Obtain the transmission and reflection coefficients at a potential step of magnitude V₀.
• 1.3 Assume a 1-dimensional well of finite depth V₀ extending from x = 0 to x = L. Find the wavefunctions and hence prove that they decay exponentially outside the well. Extend the result to three dimensions.
• 1.4 Use the WKB formula to obtain the transmission coefficient through a trapezoidal barrier.