Problem 61 - Physics for Scientists and Engineers with Modern Physics

Question

(III) Consider a particle of mass \(m\) and energy \(E\) traveling to the right where it encounters a narrow potential barrier of height \(U_{0}\) and width \(\ell\) as shown in Fig. \(38-21 .\) It can be shown that: (i) for \(EU_{0},\) the transmission probability is $$ T=\left[1+\frac{\sin ^{2}\left(G^{\prime} \ell\right)}{4\left(E / U_{0}\right)\left(E / U_{0}-1\right)}\right]^{-1} $$ where $$ G^{\prime}=\sqrt{\frac{2 m\left(E-U_{0}\right)}{\hbar^{2}}} $$ and \(R=1-T .\) Consider that the particle is an electron and it is incident on a rectangular barrier of height \(U_{0}=10 \mathrm{eV}\) and width \(\ell=1.0 \times 10^{-10} \mathrm{~m} .\) ( \(a\) ) Calculate \(T\) and \(R\) for the electron from \(E / U_{0}=0\) to \(10,\) in steps of \(0.1 .\) Make a single graph showing the two curves of \(T\) and \(R\) as a function of \(E / U_{0}\). (b) From the graph determine the energies \(\left(E / U_{0}\right)\) at which the electron will have transmission probabilities of \(10 \%\) $$ 20 \%, 50 \%, \text { and } 80 \% $$

Step-by-Step Solution

Answer

Calculate \(T\) and \(R\) using step-by-step formulas, plot them, and find \(E/U_0\) from graph for given \(T\) values.

Step 1: Understanding the Problem

We have a particle (an electron) with energy \(E\) encountering a potential barrier with height \(U_0 = 10 \text{ eV}\) and width \(\ell = 1.0 \times 10^{-10} \text{ m}\). There are two cases based on the relationship between \(E\) and \(U_0\):\(E < U_0\) and \(E > U_0\). We need to calculate the transmission \(T\) and reflection \(R = 1 - T\) probabilities across different energy ratios \( \frac{E}{U_0} \) varying from 0 to 10, in steps of 0.1.

Step 2: Setup Transmission and Reflection Formulas

For \(E < U_0\), the transmission probability is given by:\[ T = \left[1+\frac{\left(e^{G \ell}-e^{-G \ell}\right)^{2}}{16\left(\frac{E}{U_0}\right)\left(1-\frac{E}{U_0}\right)}\right]^{-1} \]For \(E > U_0\), the transmission probability becomes:\[ T = \left[1+\frac{\sin^{2}(G' \ell)}{4\left(\frac{E}{U_0}\right)\left(\frac{E}{U_0}-1\right)}\right]^{-1} \]Reflection probability for both cases is \( R = 1 - T \). We'll calculate \(G\) and \(G'\) for each case.

Step 3: Calculate \(G\) and \(G'\) Values

The formula for \(G\) when \(E < U_0\) is:\[ G = \sqrt{\frac{2m(U_0 - E)}{\hbar^2}} \]And for \(G'\) when \(E > U_0\):\[ G' = \sqrt{\frac{2m(E - U_0)}{\hbar^2}} \]Given that the particle is an electron, use its mass \(m = 9.109 \times 10^{-31}\ \text{kg}\) and \(\hbar = 1.0545718 \times 10^{-34} \text{ Js}\). We’ll calculate these for each energy ratio \(\frac{E}{U_0}\) from 0 to 10.

Step 4: Compute T and R for each \(E / U_0\) Value

For each \(E/U_0\) from 0 to 1 (exclusive), use the \(T\) formula for \(E < U_0\). For each \(E/U_0\) from 1 (inclusive) to 10, use the \(T\) formula for \(E > U_0\). Calculate \(R = 1 - T\) for each step.

Step 5: Plot T and R vs \(E / U_0\)

Using a graphing software, plot \(T\) and \(R\) values on a single graph with \(E/U_0\) on the x-axis and probabilities on the y-axis. Both \(T\) and \(R\) should be distinct curves.

Step 6: Determine Energies for Specific Transmission Probabilities

From the graph, identify the \(E/U_0\) values where \(T = 10\%, 20\%, 50\%, 80\%\). These correspond to probabilities of 0.1, 0.2, 0.5, and 0.8, respectively.

Key Concepts

Potential Barrier

Imagine an electron traveling in a straight line, moving into a region with different energy characteristics. This is known as encountering a "Potential Barrier." It's similar to how a car approaches a hill on a road. The potential barrier in physics represents a sudden shift in energy levels that affects how particles like electrons move.

In quantum mechanics, potential barriers are of particular interest. When an electron with certain energy encounters this barrier, its behavior changes based on the height and width of the barrier. The barrier height, represented as \( U_0 \), and its width, represented as \( \ell \), are critical in determining the electron's subsequent behavior.

If the electron's energy is less than the potential barrier energy, it faces difficulty surmounting it, much like a car struggling to climb a steep hill without enough speed. This analogy helps us understand why only a part of the electron's wave might "tunnel" through, a fascinating property of quantum mechanics that isn't possible in classical physics.

Transmission Probability

When we talk about transmission probability, we're interested in the likelihood that an electron will pass through a potential barrier. In quantum mechanics, even if an electron's energy is less than the height of the barrier, it can still continue its journey due to a phenomenon called "quantum tunneling."

The probability of transmission, denoted as \( T \), varies greatly with the electron's energy. If the energy \( E \) is less than the barrier's height \( U_0 \), the equation becomes a bit complicated, involving exponential terms. However, this complexity captures the unique behavior of quantum tunneling. The equation is given by:

• \( T = \left[ 1 + \frac{\left(e^{G \ell} - e^{-G \ell}\right)^2}{16 \left(\frac{E}{U_0}\right) \left(1 - \frac{E}{U_0}\right)} \right]^{-1} \)

For energies greater than \( U_0 \), the behavior shifts, and we use the sine of a term instead:

• \( T = \left[ 1 + \frac{\sin^2(G' \ell)}{4 \left(\frac{E}{U_0}\right) \left(\frac{E}{U_0} - 1\right)} \right]^{-1} \)

This elegant dependence on energy reveals how quantum mechanics allows even lower-energy particles to sneak past barriers, defying the classical expectation that higher energy is always needed to overcome obstacles.

Reflection Probability

Reflection probability is the flip side of transmission probability. When an electron encounters a potential barrier, it can either go through or bounce back. Reflection probability, denoted as \( R \), is the chance that the electron will rebound instead of continuing its path.

The intriguing aspect of reflection in quantum mechanics lies in how it's intertwined with transmission. In fact, they are complementary values:

• \( R = 1 - T \)

This simple relationship shows that if the transmission probability is high, the reflection probability is low, and vice versa.

For example, when the energy \( E \) is much lower than the potential barrier height \( U_0 \), reflection probability is high because the electron lacks the energy to breach the barrier effectively. However, due to quantum tunneling, some probability remains for it to pass through. Thus, even when reflection dominates, part of the electron's wave continues its remarkable journey.

Electron Energy

Electron energy is a key player in understanding how particles interact with potential barriers. The energy \( E \) of an electron determines whether it can overcome or bypass a barrier via tunneling.

Electron energy in this context is often measured relative to the potential barrier's height \( U_0 \). When discussing the energy ratio \( \frac{E}{U_0} \), we get insights into how easy or difficult it is for the electron to transmit through the barrier.

Consider different scenarios:

• If \( \frac{E}{U_0} < 1 \), the electron is underpowered against the barrier, making quantum tunneling the primary method to continue its path.
• If \( \frac{E}{U_0} > 1 \), classical physics would predict successful transmission, but quantum mechanics adds intriguing augmentations, like interference effects, influencing transmission probability.

Understanding electron energy dynamics enriches our grasp of quantum phenomena, especially how particles smoothly tunnel through barriers seemingly without sufficient classical energy.

Quantum Mechanics

Quantum mechanics is the grand framework that allows us to explore phenomena at very tiny scales, where conventional physics doesn't suffice. It's the realm where particles like electrons behave in unimaginable ways compared to large-scale objects.

At the core of quantum mechanics is the idea that particles exhibit both particle and wave-like properties. This dual nature is crucial in explaining phenomena like quantum tunneling. When electrons encounter potential barriers, instead of stopping completely, they have a probability to transition through,

• Defying classical expectations, it's as if traffic passes through a roadblock without breaking it.

Quantum mechanics introduces the concept that at tiny scales, the exact position and velocity of a particle can't be known simultaneously with precision — a concept known as the "Heisenberg Uncertainty Principle."

In addition, wave functions and principles like superposition and entanglement expose a captivating vista of physics that challenges our intuition but beautifully explains the intricate workings of the universe at its fundamental level.