Step Potential and Barriers: Quantum Reflection and Transmission

Table of Contents

1. Introduction
2. Classical vs Quantum Behavior
3. Step Potential Definition
4. Regions and Wavefunction Forms
5. Solving the Schrödinger Equation
6. Case 1: \( E > V_0 \) — Partial Transmission
7. Case 2: \( E < V_0 \) — Tunneling and Reflection
8. Reflection and Transmission Coefficients
9. Probability Current and Conservation
10. Physical Interpretation
11. Sharp vs Gradual Step Potentials
12. Quantum vs Classical Reflection
13. Time-Dependent Considerations
14. Connection to Barrier Potentials
15. Applications in Devices and Quantum Optics
16. Conclusion

1. Introduction

The step potential is one of the simplest and most instructive examples in quantum mechanics. It illustrates fundamental principles of quantum reflection, transmission, and tunneling. Unlike classical mechanics, quantum particles can reflect even when they have enough energy to surpass a potential step.

2. Classical vs Quantum Behavior

In classical physics:

• \( E > V_0 \): full transmission
• \( E < V_0 \): total reflection

Quantum mechanics changes this:

• Wave nature leads to partial reflection even when \( E > V_0 \)
• Tunneling occurs when \( E < V_0 \)

3. Step Potential Definition

\[
V(x) = \begin{cases}
0, & x < 0 \
V_0, & x \geq 0
\end{cases}
\]

This introduces a discontinuity in potential energy at \( x = 0 \).

4. Regions and Wavefunction Forms

Let total energy \( E \) and particle mass \( m \).

Region I ( \( x < 0 \) ):

\[
\psi_I(x) = Ae^{ikx} + Be^{-ikx}, \quad k = \frac{\sqrt{2mE}}{\hbar}
\]

Region II ( \( x > 0 \) ):

• If \( E > V_0 \):
  \[
  \psi_{II}(x) = Ce^{iqx}, \quad q = \frac{\sqrt{2m(E – V_0)}}{\hbar}
  \]
• If \( E < V_0 \):
  \[
  \psi_{II}(x) = Ce^{-\kappa x}, \quad \kappa = \frac{\sqrt{2m(V_0 – E)}}{\hbar}
  \]

5. Solving the Schrödinger Equation

Apply boundary conditions at \( x = 0 \):

• Continuity of \( \psi(x) \)
• Continuity of \( \frac{d\psi}{dx} \)

These yield equations for reflection and transmission amplitudes.

6. Case 1: \( E > V_0 \) — Partial Transmission

Even though the particle has enough energy:

• Part of the wave reflects back due to the discontinuity
• The rest transmits with modified wavelength

Coefficients:

\[
R = \left| \frac{k – q}{k + q} \right|^2, \quad T = \frac{4kq}{(k + q)^2}
\]

Where:

• \( R \): reflection coefficient
• \( T \): transmission coefficient
• \( R + T = 1 \)

7. Case 2: \( E < V_0 \) — Tunneling and Reflection

• Classically forbidden region \( x > 0 \)
• Wavefunction decays exponentially in Region II
• Particle has a finite probability to be found in \( x > 0 \), but cannot transmit infinitely

Reflection coefficient:

\[
R = 1
\]

Transmission coefficient \( T = 0 \), but penetration still occurs.

8. Reflection and Transmission Coefficients

Define current densities:

\[
j = \frac{\hbar}{2mi} \left( \psi^* \frac{d\psi}{dx} – \psi \frac{d\psi^*}{dx} \right)
\]

• Incident current: \( j_{\text{inc}} \propto |A|^2 \)
• Reflected current: \( j_{\text{ref}} \propto |B|^2 \)
• Transmitted current: \( j_{\text{trans}} \propto |C|^2 \)

Use these to derive \( R \) and \( T \).

9. Probability Current and Conservation

Quantum mechanics conserves probability:

\[
R + T = 1
\]

Confirms that all probability is accounted for — either reflected or transmitted.

10. Physical Interpretation

• Reflection arises due to wavefunction discontinuity in slope, even when \( E > V_0 \)
• Tunneling into forbidden region occurs even if no classical path exists
• Highlights core principles of quantum superposition and boundary-driven dynamics

11. Sharp vs Gradual Step Potentials

• Ideal step: discontinuous \( V(x) \)
• Smooth step: \( V(x) \) varies continuously (e.g., hyperbolic tangent)

Gradual steps result in reduced reflection, used in quantum well and heterojunction modeling.

12. Quantum vs Classical Reflection

Quantum mechanics predicts behavior impossible classically.

13. Time-Dependent Considerations

Wave packets offer richer dynamics:

• Partial packet reflects
• Remainder transmits or tunnels
• Leads to interference and spread

Important for real experiments and ultrafast electronics.

14. Connection to Barrier Potentials

Step potentials are building blocks for:

• Finite barriers
• Potential wells
• Quantum dots
• Semiconductor band structures

Understanding the step potential enables modeling complex layered structures.

15. Applications in Devices and Quantum Optics

• Photodiodes and tunnel junctions
• Quantum well lasers
• Carrier injection modeling in semiconductors
• Quantum reflection used in atom mirrors and optical traps

16. Conclusion

The step potential illustrates the fundamental differences between classical and quantum behavior when encountering energy discontinuities. It showcases partial reflection, tunneling, and wave-like interference. This simple model forms the basis for understanding more complex quantum barriers, wells, and layered systems across physics, electronics, and optics.