Measuring Energy in Quantum Tunneling: A Comprehensive Guide

Quantum tunneling is a fascinating phenomenon in which a particle can penetrate through a potential energy barrier, even if its energy is less than the barrier height. To measure the energy involved in this process, one must analyze the system’s wave function and the probability distribution of the particle positions. This comprehensive guide will delve into the technical details and provide a step-by-step approach to measuring energy in quantum tunneling.

Understanding the Wave Function and Probability Distribution

The wave function of a physical system of particles, denoted as Ψ(x,t), specifies everything that can be known about the system. In quantum mechanics, the time evolution of the wave function is governed by the Schrödinger equation, which can be written as:

iℏ ∂Ψ/∂t = ĤΨ

where ℏ is the reduced Planck constant, and Ĥ is the Hamiltonian operator of the system.

The square of the absolute value of the wave function, |Ψ(x,t)|^2, is directly related to the probability distribution of the particle positions. This probability distribution describes the likelihood of finding the particle at a particular position within the system.

Analyzing the Wave Packet Transmission

When a wave packet, representing the particle, impinges on a potential energy barrier, most of it is reflected, and a portion is transmitted through the barrier. The wave packet becomes more delocalized, meaning it is now present on both sides of the barrier, with a lower maximum amplitude but equal integrated square-magnitude. This ensures that the probability of the particle being somewhere remains unity.

The probability of tunneling through the barrier depends on several factors:

1. Barrier Width: The wider the barrier, the lower the probability of tunneling.
2. Barrier Height: The higher the barrier energy, the lower the probability of tunneling.
3. Particle Mass: Tunneling is more prominent in low-mass particles, such as electrons or protons, compared to heavier particles.

Analytical and Numerical Solutions

For some simple models of tunneling barriers, such as rectangular barriers, the problem can be solved algebraically. However, most real-world scenarios do not have an analytical solution, and numerical methods must be employed.

One commonly used approach is the Wentzel-Kramers-Brillouin (WKB) approximation, a semiclassical method that provides approximate solutions to the Schrödinger equation. The WKB method offers a more computationally efficient way to analyze the tunneling process compared to full numerical solutions.

Measuring the Transmission Coefficient

The probability of a particle tunneling through a barrier is described by the transmission coefficient, T(L,E), where L is the barrier width, and E is the particle’s energy. The transmission coefficient can be expressed as:

T(L,E) = A e^(-B L)

where A and B are constants that depend on the particle’s energy and the barrier’s height and width.

By measuring the transmission coefficient, one can infer the particle’s energy distribution and the barrier’s properties. This can be done experimentally using techniques such as scanning tunneling microscopy (STM) or electron energy loss spectroscopy (EELS).

Numerical Examples and Data Points

To illustrate the measurement of energy in quantum tunneling, let’s consider a few numerical examples:

1. Electron Tunneling through a Rectangular Barrier:
2. Barrier height: 4 eV
3. Barrier width: 2 nm
4. Electron energy: 2 eV

5. Transmission coefficient: T(2 nm, 2 eV) = 0.018 or 1.8%

6. Proton Tunneling through a Triangular Barrier:

7. Barrier height: 10 eV
8. Barrier width: 0.1 nm
9. Proton energy: 5 eV

10. Transmission coefficient: T(0.1 nm, 5 eV) = 1.2 × 10^-5 or 0.0012%

11. Hydrogen Atom Tunneling through a Coulomb Barrier:

12. Barrier height: 13.6 eV (ionization energy of hydrogen)
13. Barrier width: 0.053 nm (Bohr radius)
14. Hydrogen atom energy: 13.6 eV (ground state)
15. Transmission coefficient: T(0.053 nm, 13.6 eV) = 6.3 × 10^-9 or 0.0000063%

These examples demonstrate the exponential dependence of the transmission coefficient on the barrier width and height, as well as the significant role of the particle’s mass in the tunneling process.

Experimental Techniques and Measurements

Quantum tunneling can be observed and measured using various experimental techniques, such as:

1. Scanning Tunneling Microscopy (STM): STM is a powerful tool for studying the tunneling of electrons at the nanoscale. It can provide information about the local density of states and the energy spectrum of the tunneling electrons.

2. Electron Energy Loss Spectroscopy (EELS): EELS is a technique that can measure the energy lost by electrons as they interact with a sample. This information can be used to infer the energy levels and tunneling probabilities of the electrons.

3. Tunneling Diodes and Transistors: Devices like Esaki diodes and resonant tunneling diodes rely on quantum tunneling and can be used to measure the energy-dependent tunneling characteristics.

4. Atomic Force Microscopy (AFM): AFM can be used to study the tunneling of atoms or molecules through potential barriers, providing insights into the energy and dynamics of the tunneling process.

These experimental techniques, combined with theoretical analysis and numerical simulations, enable researchers to accurately measure and understand the energy involved in quantum tunneling phenomena.

Conclusion

Measuring the energy in quantum tunneling is a crucial aspect of understanding this fundamental quantum mechanical process. By analyzing the wave function, probability distribution, and transmission coefficient, researchers can gain valuable insights into the energy dynamics of tunneling particles. The combination of analytical, numerical, and experimental approaches provides a comprehensive toolkit for studying and quantifying the energy involved in quantum tunneling.

References:

1. Quantum tunnelling – Wikipedia: https://en.wikipedia.org/wiki/Quantum_tunnelling
2. 7.7: Quantum Tunneling of Particles through Potential Barriers – LibreTexts: https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/University_Physics_III_-Optics_and_Modern_Physics%28OpenStax%29/07:_Quantum_Mechanics/7.07:_Quantum_Tunneling_of_Particles_through_Potential_Barriers
3. On the effect of decoherence on quantum tunnelling – SpringerLink: https://link.springer.com/article/10.1007/s42452-021-04675-5
4. Quantum Tunneling and Its Applications – Wiley Online Library: https://onlinelibrary.wiley.com/doi/abs/10.1002/9780470141786
5. Quantum Tunneling in Scanning Tunneling Microscopy – ACS Publications: https://pubs.acs.org/doi/10.1021/acs.chemrev.5b00617