Consider A Rectangular Potential Barrier: 0 Otherwise. With Vo > 0 And A > 0. Show That The Transmission
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Introduction to Quantum Tunneling and Potential Barriers
Quantum mechanics introduces phenomena that defy classical intuition, one of which is the concept of quantum tunneling. In classical physics, a particle with energy less than a potential barrier is reflected completely; it cannot penetrate or pass through the barrier. However, quantum particles are described by wave functions that can extend into classically forbidden regions, making it possible, with some probability, to tunnel through barriers even when their energy is less than the barrier height.
This phenomenon has profound implications in various fields, including nuclear physics, semiconductor physics, and quantum computing. To analyze tunneling quantitatively, physicists typically model barriers as potential energy functions and solve the Schrödinger equation to obtain transmission and reflection coefficients.
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Modeling the Rectangular Potential Barrier
The Potential Profile
The potential barrier under consideration is modeled as a rectangular barrier:
\[
V(x) =
\begin{cases}
V_0, & \text{for } 0 \leq x \leq A \\
0, & \text{otherwise}
\end{cases}
\]
where:
- \( V_0 > 0 \) is the height of the barrier.
- \( A > 0 \) is the width of the barrier.
This simple model captures the essence of tunneling phenomena and allows for analytical solutions to the Schrödinger equation.
Wave Function Regions
We consider a particle with energy \( E \) approaching the barrier from the left. The wave functions in different regions are:
- Region I (\( x < 0 \)): Incident wave and reflected wave
- Region II (\( 0 \leq x \leq A \)): Wave within the barrier
- Region III (\( x > A \)): Transmitted wave
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Solving the Schrödinger Equation
The Schrödinger Equation
For a particle of mass \( m \), the time-independent Schrödinger equation is:
\[
-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E \psi(x)
\]
where \( \hbar \) is the reduced Planck's constant.
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Wave Function Solutions in Different Regions
Region I (\( x < 0 \)):
\[
\psi_I(x) = Ae^{ikx} + Be^{-ikx}
\]
where:
\[
k = \frac{\sqrt{2mE}}{\hbar}
\]
- \( Ae^{ikx} \): incident wave
- \( Be^{-ikx} \): reflected wave
Region II (\( 0 \leq x \leq A \)):
\[
\psi_{II}(x) = Ce^{iqx} + De^{-iqx}
\]
where:
\[
q = \frac{\sqrt{2m(V_0 - E)}}{\hbar}
\]
Note: Since \( E < V_0 \), \( q \) is real and positive, indicating an exponentially decaying wave inside the barrier.
Region III (\( x > A \)):
\[
\psi_{III}(x) = Fe^{ikx}
\]
representing the transmitted wave traveling to the right.
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Applying Boundary Conditions
To determine the transmission coefficient, we need to match the wave functions and their derivatives at the boundaries \( x=0 \) and \( x=A \):
- Continuity of \( \psi \):
\psiI(0) = \psi{II}(0), \quad \psi{II}(A) = \psi{III}(A)
\]
- Continuity of \( d\psi/dx \):
\frac{d\psiI}{dx}\bigg|{x=0} = \frac{d\psi{II}}{dx}\bigg|{x=0}, \quad \frac{d\psi{II}}{dx}\bigg|{x=A} = \frac{d\psi{III}}{dx}\bigg|{x=A}
\]
By applying these boundary conditions, we derive relationships between the coefficients \( A, B, C, D, F \).
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Deriving the Transmission Coefficient
Expression for Transmission Coefficient \( T \)
The key quantity of interest is the transmission coefficient \( T \), defined as the ratio of transmitted flux to incident flux:
\[
T = \frac{\text{transmitted current density}}{\text{incident current density}} = \left| \frac{F}{A} \right|^2 \frac{k}{k} = \left| \frac{F}{A} \right|^2
\]
The detailed derivation yields the well-known expression:
\[
T = \frac{1}{1 + \frac{V0^2 \sinh^2(qA)}{4E(V0 - E)}}
\]
which simplifies to:
\[
T = \frac{1}{1 + \frac{V0^2}{4E(V0 - E)} \sinh^2(qA)}
\]
This formula indicates that, even for \( E < V_0 \), the transmission probability is non-zero, illustrating quantum tunneling.
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Analysis of the Transmission Coefficient
Behavior for Different Regimes
- Thin and Low Barriers (\( A \to 0 \), \( V_0 \to 0 \)):
Almost complete transmission, as expected classically.
- High and Wide Barriers (\( V_0 \gg E \), \( A \to \infty \)):
\[
T \to 0
\]
Very low probability of tunneling, consistent with classical intuition.
- Intermediate Regimes:
The tunneling probability depends exponentially on the width \( A \) and the barrier height \( V_0 \), governed mainly by the hyperbolic sine term \( \sinh^2(qA) \).
Approximate Expression for Large \( qA \)
When \( qA \gg 1 \), \( \sinh(qA) \approx \frac{1}{2} e^{qA} \), leading to:
\[
T \approx \frac{16E(V0 - E)}{V0^2} e^{-2qA}
\]
This exponential decay emphasizes the sensitivity of tunneling probability to barrier parameters.
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Physical Interpretation and Significance
The derivation of the transmission coefficient from the rectangular barrier model reveals that quantum particles can penetrate barriers with finite probability, a phenomenon absent in classical physics. The probability diminishes exponentially with increasing barrier width and height but remains nonzero, enabling phenomena such as nuclear fusion in stars and operation of tunnel diodes.
Furthermore, understanding the dependence of \( T \) on parameters \( V_0 \), \( A \), and \( E \) enables engineers and physicists to design quantum devices with desired tunneling characteristics.
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Conclusion
By modeling the potential barrier as a rectangular step and solving the Schrödinger equation with appropriate boundary conditions, we derive an explicit expression for the transmission coefficient \( T \). This derivation confirms that even when \( E < V0 \), quantum particles possess a finite probability of tunneling through the barrier. The key factors influencing tunneling probability include the barrier height \( V0 \), width \( A \), and the particle's energy \( E \).
The analysis underscores the fundamental departure of quantum mechanics from classical intuition and provides the mathematical foundation for numerous technological innovations such as tunnel diodes, scanning tunneling microscopes, and quantum computing components.
