A particle with mass m and total energy E tunnels through a...

A particle with mass $m$ and total energy $E$ tunnels through a square barrier of height $U_{0}$ and width $L .$ When the transmission coefficient is not much less than unity, it is given by $$ T=\left[1+\frac{\left(U_{0} \sinh \kappa L\right)^{2}}{4 E\left(U_{0}-E\right)}\right]^{-1} $$ where $\sinh \kappa L=\left(e^{\kappa L}-e^{-\kappa L}\right) / 2$ is the hyperbolic sine of $\kappa L$ (a) Show that if $\kappa L \gg 1,$ this expression for $T$ approaches Eq. $(40.21) .$ (b) Explain why the restriction $\kappa L \gg 1$ in part (a) implies either that the barrier is relatively wide or that the energy $E$ is relatively low compared to $U_{0}$ . (c) Show that as the particle's incident kinetic energy $E$ approaches the barrier height $U_{0}, T$ approaches $\left[1+(k L / 2)^{2}\right]^{-1},$ where $k=\sqrt{2 m E} / \hbar$ is the wave number of the incident particle. (Hint If $|z| \ll 1,$ then $\sinh z \approx z . )$

A particle with mass $m$ and total energy $E$ tunnels through a square barrier of height $U_{0}$ and width $L .$ When the transmission coefficient is not much less than unity, it is given by
$$
T=\left[1+\frac{\left(U_{0} \sinh \kappa L\right)^{2}}{4 E\left(U_{0}-E\right)}\right]^{-1}
$$
where $\sinh \kappa L=\left(e^{\kappa L}-e^{-\kappa L}\right) / 2$ is the hyperbolic sine of $\kappa L$
(a) Show that if $\kappa L \gg 1,$ this expression for $T$ approaches Eq. $(40.21) .$ (b) Explain why the restriction $\kappa L \gg 1$ in part (a) implies either that the barrier is relatively wide or that the energy $E$ is relatively low compared to $U_{0}$ . (c) Show that as the particle's incident kinetic energy $E$ approaches the barrier height $U_{0}, T$ approaches $\left[1+(k L / 2)^{2}\right]^{-1},$ where $k=\sqrt{2 m E} / \hbar$ is the wave number of the incident particle. (Hint If $|z| \ll 1,$ then $\sinh z \approx z . )$

Key Concepts

Quantum Tunneling

Quantum tunneling is the phenomenon where a particle has a nonzero probability of passing through a potential barrier, even when its energy is less than the height of the barrier. This behavior is a direct consequence of the wave nature of particles in quantum mechanics and stands in contrast to classical physics, where a particle would be completely reflected by such a barrier.

Transmission Coefficient

The transmission coefficient is a measure of the probability amplitude that a particle will tunnel through a potential barrier. It is derived from solving the Schrödinger equation in the presence of the barrier and quantifies how the particle’s wave function decays inside the barrier relative to its incident amplitude.

Square Potential Barrier

A square potential barrier is a simplified model in quantum mechanics where the potential energy is constant over a finite region of space and zero elsewhere. Its analytical tractability makes it a common example for studying tunneling phenomena and understanding the fundamental aspects of quantum barrier penetration.

Hyperbolic Functions and Their Approximations

Hyperbolic functions, such as the hyperbolic sine (sinh), appear in the solutions of the Schrödinger equation for a constant potential barrier. Their properties, especially their behavior in the limits of large or small arguments, are crucial for approximating the transmission coefficient under different barrier conditions, such as when the barrier is very wide or when the comparing energy is close to the barrier height.

Energy Dependence and Barrier Parameters

The tunneling probability depends critically on the relationship between the particle's energy and the barrier height as well as on the barrier width. In particular, the decay constant ?, which appears in the expression for the transmission coefficient, is determined by the energy difference between the barrier height and the particle's energy. This relationship explains why a wide barrier or a low particle energy relative to the barrier height leads to a rapid decay of the wave function inside the barrier, thereby reducing the tunneling probability.

Transcript

00:01 In this exercise, we have a more general transmission probability, t that's given here on the first line, by 1 plus u0 times the hyperbolic sign of kl squared, divided by 4eu0 minus e to the minus 1, where u0 is the height of the potential barrier, e is the energy of the particle, l is the length of the potential barrier, and k is equal to the square root of 2m times e.

00:31 U0 minus e divided by h bar.

00:35 In question a, we have to show that when kl is much greater than 1, the transmission probability reduces to the formula that we had before in the book.

00:50 That's the second equation here on the screen.

00:54 T is equal to 16e divided by u0 times 1 minus e divided by u0 times e to minus 2, kl.

01:01 So basically i'm going to take the first equation, and then i'm going to impose that kl is much greater than one.

01:11 First, notice that the hyperbolic sign of kl is equal to one -half of e to the kl minus -kl minus -k -l, and when kl is much greater than 1, e to the minus kl tends to 0, so this is approximately one -half of e to the k -kl.

01:32 So you can go back to the transmission, which is 1 plus u0 times the sign, the hyperbolic sign of kl, which as i just showed, is approximately e to the kl divided by 2 squared divided by 16eu0 minus e.

02:10 So this is, i'm sorry, it's not 16, this should be 4.

02:17 So this is equal to 1 plus u0 squared, e to the 2kl divided by 16, e u0 minus e to the minus 1.

02:34 Now notice that if kl is much greater than 1, then we have that the exponential of 2kl is much greater than 1, 1.

02:47 So we can disregard the first term that is this one here.

02:52 I'm simply going to disregard it.

02:55 So t is equal to is approximately equal to u0 squared e to 2kl divided by 16e u0 minus e to the minus 1.

03:14 And this is e to the minus 2kl times 16 e u0 minus e divided by u0 squared so this is equal to 16 e divided by u0 times 1 minus e divided by u 0 times e to the minus 2 kl okay so this is exactly the expression that we had the book for kl much wider than one.

03:54 Then we have to explain why imposing that kl is much greater than one implies that either the barrier is relatively wide or that the energy e of the particle is small compared with u0.

04:17 Well, notice that for k1, l to be much greater than one, either l is much greater than 1 over k, okay...

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