L16.3 The delta function potential: reflection and transmission coefficients derivation

Summary

The video script discusses solving equations related to wave reflection and transmission coefficients. The speaker explains how the number of equations and unknowns should match to solve the system, then uses algebraic manipulations to reduce unknowns. They derive formulas for the reflection and transmission coefficients in terms of beta, related to energy and mass. The coefficients help predict the probability of wave transmission and reflection based on incident amplitude. The conclusion links the energy of waves to their reflection and transmission, aligning with theoretical expectations in wave mechanics.

Takeaways

• 🧮 The equation starts as A + B = F + G, and G is later set to zero, simplifying it further.
• 🔄 Manipulation of equations leads to a result where B is expressed in terms of I (imaginary unit) and beta.
• 📉 The reflected wave amplitude (B) is related to the incident wave amplitude (A) through the equation B = I * beta * A.
• 🌊 The transmitted amplitude (F) is expressed in terms of the incident amplitude (A), simplifying as F = A / (1 - I * beta).
• 📊 The reflection coefficient (R) is calculated using the squared ratio of reflected to incident amplitudes.
• 🔗 The reflection coefficient (R) is found to be proportional to beta squared divided by 1 plus beta squared.
• 📡 The transmission coefficient (T) is computed similarly and is inversely proportional to 1 plus beta squared.
• ⚖️ It's confirmed that the sum of the reflection and transmission coefficients equals 1, ensuring energy conservation in the system.
• 🔄 Beta (β) is defined as M * alpha / (h-bar squared), which is related to the reflection and transmission coefficients.
• 💡 Higher energy results in increased transmission, whereas lower energy leads to more reflection, aligning with physical expectations.

Q & A

What is the primary equation discussed in the transcript?

The primary equation discussed is A + B = F + G, which simplifies to A + B = F and G = 0.

Why is G set equal to zero in the equation?

G is set to zero to simplify the system of equations, allowing the focus to be on the relationship between A, B, and F.

What challenge is faced with the number of unknowns and equations?

The challenge is that there are three unknowns (A, B, F) but only two equations, making it impossible to solve the system directly.

How is the problem solved despite the mismatch between equations and unknowns?

The problem is solved by reducing the number of unknowns to two by expressing A in terms of F, which allows the two equations to be solved.

What role does 'beta' play in the equations?

'Beta' is introduced as a simplification factor, defined as m alpha / h bar squared, to make the equations more manageable.

What is the amplitude of the reflected wave in terms of the incident wave?

The amplitude of the reflected wave, V, is given as V = I * beta / (1 - I * beta) * A.

What are the reflection and transmission coefficients?

The reflection coefficient is R = beta^2 / (1 + beta^2), and the transmission coefficient is T = 1 / (1 + beta^2).

How are the reflection and transmission coefficients related to energy?

The reflection coefficient is inversely proportional to energy, while the transmission coefficient is directly proportional to energy.

What conclusion is drawn regarding energy and wave behavior?

As the energy of the wave increases, the transmission increases, while reflection decreases. This is consistent with the physical expectation that higher energy waves are more likely to transmit.

What is the significance of proving that R + T = 1?

Proving that R + T = 1 ensures conservation of probability, meaning that all of the incident wave is accounted for in either reflection or transmission, with no absorption.