L14.1 Gauge invariance of the Schrödinger Equation

TL;DR

Electromagnetic fields and quantum mechanics are studied in relation to particles, revealing significant changes to the Schrodinger equation and the Hamiltonian.

Transcript

PROFESSOR: So we continue today our study of electromagnetic fields, and quantum mechanics, and particles in those electromagnetic fields. So last time, we described what we must do in order to couple a particle to an electromagnetic field. And the rule was oddly simple. You get the Schrodinger equation, and the Hamiltonian is now changed into this... Read More

Key Insights

• 🥺 Coupling a particle to electromagnetic fields leads to changes in the Schrodinger equation and Hamiltonian, introducing a new term involving the vector potential A.
• 💱 Electromagnetic potentials can be transformed through gauge transformations without changing the physical properties of the system.
• 🤩 The gradient operator plays a key role in the Hamiltonian, representing the derivative of the vector potential with respect to position.
• 🛀 The identity involving the gauge covariant derivative simplifies the proof of gauge invariance, showing that the derivative operator is compatible with gauge transformations.
• 💆 The canonical momentum, P, is not simply mass times velocity, but rather a momentum that generates translation in the system.

Questions & Answers

Q: How does coupling a particle to electromagnetic fields affect the Schrodinger equation?

The Schrodinger equation with the new gauge potentials, A prime and phi prime, is not solved by the same wave function as the Schrodinger equation with the old potentials, A and phi. The wave function must change, and the relationship between the two wave functions is given by a function U involving the gauge parameter and physical constants.

Q: What is the significance of gauge invariance in relation to the Hamiltonian?

Gauge invariance ensures that the physics remains invariant under gauge transformations. The Hamiltonian, with terms involving the vector potential A, accounts for this by demonstrating how a change in gauge potential affects the entire wave function, not just the potential itself.

Q: What is the role of the gradient operator in the Hamiltonian?

The gradient operator, acting on the vector potential A, represents the derivative of A with respect to position. It plays a crucial role in the Hamiltonian, particularly in the term involving the dot product between the momentum operator P and A. The term P dot A must be understood as the gradient of A acting on the wave function, accounting for the interaction between the momentum and the vector potential.

Q: How does the identity involving the gauge covariant derivative contribute to gauge invariance?

The identity, derived using the gauge covariant derivative, demonstrates that the derivative operator can be moved across the gauge transformation function U, effectively transforming the gauge potential from A prime to A. This property of the derivative is crucial for establishing gauge invariance and proving that the Schrodinger equation remains invariant under gauge transformations.

Summary & Key Takeaways

• The Schrodinger equation and Hamiltonian change when coupling a particle to electromagnetic fields, with the addition of a new term involving the vector potential A.

• Electromagnetic potentials can be transformed through gauge transformations, making them physically equivalent.

• Gauge invariance requires the wave function to change when transforming between different gauge potentials.