Problem 63 - University Physics with Modern Physics

Question

A particle with mass \(m\) and total energy \(E\) tunnels through a square barrier of height \(U_{0}\) and width \(L .\) When the trans- mission 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. \((40.42) .\) (b) Explain why the restriction \(\kappa L>1\) in part (a) implies either that the barrier is relatively wide or 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 \(|z|<1,\) then \(\sinh z \approx z . )\)

Step-by-Step Solution

Answer

For \(\kappa L \gg 1\), T approaches the form due to exponential dominance. \(\kappa L > 1\) implies a wide barrier or low \(E\). As \(E \to U_0\), T approaches \(\left[1+(kL/2)^2\right]^{-1}\).

Step 1: Understanding the Problem

We have a particle with mass \(m\) and total energy \(E\) tunneling through a potential barrier of height \(U_0\) and width \(L\). The transmission coefficient \(T\) is given by a formula involving hyperbolic sine functions. We need to show approximations of this formula under certain conditions, like \(\kappa L \gg 1\) and \(E \approx U_0\).

Step 2: Defining Key Variables

The hyperbolic sine function \(\sinh \kappa L\) is defined as \(\frac{e^{\kappa L} - e^{-\kappa L}}{2}\), where \(\kappa = \sqrt{\frac{2m(U_0 - E)}{\hbar^2}}\). The condition \(\kappa L \gg 1\) implies that the exponential terms will dominate, allowing simplifications.

Step 3: Simplifying Transmission Coefficient for \(\kappa L \gg 1\)

When \(\kappa L \gg 1\), \(\sinh \kappa L \approx \frac{1}{2} e^{\kappa L}\). Substituting this into the transmission coefficient: \[T = \left[1 + \frac{(U_0 \cdot \frac{1}{2} e^{\kappa L})^2}{4E(U_0 - E)}\right]^{-1} \approx \left[1 + \frac{U_0^2 \cdot \frac{1}{4} e^{2\kappa L}}{4E(U_0 - E)}\right]^{-1}\]This simplifies further and can approach the form required in the question as \(\kappa L\) becomes large.

Step 4: Analyzing Condition \(\kappa L > 1\)

The condition \(\kappa L > 1\) indicates either a wide barrier (large \(L\)) or low particle energy \(E\) relative to \(U_0\). This is because \(\kappa\) is proportional to the square root of \(U_0 - E\), so increases in \(L\) or decreases in \(E\) relative to \(U_0\) increase \(\kappa L\).

Step 5: Approximating \(T\) as \(E \to U_0\)

If \(E \to U_0\), then \(\kappa \to 0\). For small \(\kappa L\), \(\sinh \kappa L \approx \kappa L\). Substituting into the formula: \[T = \left[1 + \frac{(U_0 \cdot \kappa L)^2}{4E(U_0 - E)}\right]^{-1} \approx \left[1 + \frac{(kL/2)^2}{1}\right]^{-1}\]This matches the form given by the question, \(\left[1 + (kL/2)^2\right]^{-1}\), as expected.

Key Concepts

transmission coefficient

The transmission coefficient, represented by the symbol \( T \), is a key concept in quantum tunneling. It quantifies the probability that a particle will successfully tunnel through a potential barrier. This is not a common phenomenon in classical physics, where particles lack enough energy to overcome the barrier. However, in quantum mechanics, particles can behave like waves, allowing them to pass through barriers with energies lower than their own.
The formula for the transmission coefficient can vary under different conditions, but in this scenario, it is given by:

• \[T = \left[1 + \frac{(U_0 \sinh \kappa L)^2}{4E(U_0 - E)}\right]^{-1}\]

This expression helps in understanding how varying the potential barrier or the energy of the particle affects the likelihood of successful tunneling.
Under certain conditions, like \( \kappa L \gg 1 \), the transmission coefficient can be simplified, reflecting a lesser probability for particles to pass through very thick barriers. This provides insight into the delicate balance between particle energy, barrier height, and width in determining tunneling behavior.

potential barrier

A potential barrier in quantum mechanics acts as an obstacle to particle motion. It is characterized by its height \( U_0 \) and width \( L \). This kind of barrier is central to numerous quantum phenomena, including tunneling, which occurs when a particle passes through a barrier it would not normally be able to in classical terms.
For a particle approaching such a barrier, its behavior is dictated by the relative sizes of its energy \( E \) and the barrier height \( U_0 \).

• If \( E \) is greater than \( U_0 \), the particle can move over the barrier.
• If \( E \) is less than \( U_0 \), the particle must rely on tunneling.

In quantum tunneling, the limitation imposed by a potential barrier isn't absolute. This scenario is often described by operators defining wave functions that extend into the barrier, allowing non-zero probability of finding the particle on the other side, despite not having enough energy to "climb" the barrier in the classical sense.

hyperbolic sine function

The hyperbolic sine function (\( \sinh \)) emerges in the context of quantum tunneling through the calculation of the transmission coefficient. It is part of the expression involving \( \kappa L \), noted in the formula \( \sinh \kappa L = \frac{e^{\kappa L} - e^{-\kappa L}}{2} \).
This function becomes significant particularly when \( \kappa L \) is large, (

• If \( \kappa L \gg 1 \), \( \sinh \kappa L \) simplifies to \( \frac{1}{2} e^{\kappa L} \).

Such simplification underscores the prominence of exponential terms when evaluating large arguments of hyperbolic functions.
One might wonder why \( \sinh \) instead of regular sine is used. This lies in the mathematics governing quantum mechanics, where certain differential equations naturally lead to hyperbolic rather than trigonometric functions. Trigonometric functions like sine and cosine are used for oscillatory behavior, whereas hyperbolic functions are suited for exponential-type decay or growth, typical of quantum tunneling scenarios.