Probability current and current conservation

TL;DR

The real computation involves calculating the right hand side of the equation, which leads to the discovery of current density and current conservation.

Transcript

BARTON ZWIEBACH: After this long detour, you must think that one is just trying to avoid doing the real computation, so here comes, the real computation. The real computation is taking that right hand side on the top of the blackboard and trying to just calculate this right hand side. So back to the calculation. The calculation dN/dt is equal to th... Read More

Key Insights

• 🫱 The real computation involves calculating the right hand side of the equation and leads to the discovery of the current density.
• ☠️ Current conservation is established through the equation d-rho/dt + dJ/dx = 0, where dJ/dx represents the rate of change of the current density.
• ⏳ The current density represents the flow of probability per unit time in the system.
• 🍉 The cancellation of certain terms in the equation simplifies the derivation and highlights the unique properties of the system.
• 🇦🇪 The units of the current density indicate that it represents probability per unit time.
• 🦾 The discovery of the current density expands our understanding of quantum mechanics and its similarities with electromagnetism.
• 👋 The calculation of the right hand side allows us to satisfy the conditions for the wave function, such as it approaching zero as x goes to infinity.

Questions & Answers

Q: What is the purpose of calculating the right hand side of the equation?

By calculating the right hand side, we can determine the behavior of the current density and establish current conservation in the system.

Q: Why do the terms related to the Hamiltonian and potential cancel each other out?

The cancellation of these terms is due to the opposite sign between the two terms, resulting in a neat simplification of the equation.

Q: How is the concept of total derivative used in this derivation?

By showing that the integrand is a total derivative with respect to x, the integral can be evaluated at the boundaries, allowing for the possibility of the integral to be equal to zero.

Q: What is the significance of the current density in this context?

The current density plays a crucial role in understanding current conservation and provides a deeper insight into the behavior of the system.

Summary & Key Takeaways

• The focus shifts to calculating the right hand side of the equation, which involves terms related to the Hamiltonian and the potential.

• By simplifying the equation, it is revealed that the quantity d-rho/dt can be written as a total derivative with respect to x.

• The derivation leads to the identification of the current density, which is a key concept in current conservation.