# Lecture 4 | Modern Physics: Quantum Mechanics (Stanford) | Summary and Q&A

## Summary

This video discusses the concepts of discrete and continuous variables in quantum mechanics. It explains the rules of probability in quantum mechanics, including the relationship between probabilities and amplitudes. It also introduces the concept of the delta function and its role in calculations with continuous variables. The video then explores the inner product structure for discrete and continuous variables, as well as the notion of periodicity in wave functions. Finally, it discusses the quantization of momentum and angular momentum in quantum mechanics.

## Questions & Answers

### Q: What is the first rule of probability in quantum mechanics?

The sum of probabilities for all possible outcomes should add up to one.

### Q: How are probabilities related to amplitudes?

Probabilities are governed by amplitudes, which are related to the inner product of state vectors with other state vectors.

### Q: How are basis vectors associated with discrete variables represented?

Basis vectors associated with discrete variables can be taken to form an orthonormal basis, with each vector having a length of one and being mutually orthogonal.

### Q: How can the delta function be thought of in terms of discrete variables?

The delta function can be thought of as a function of the difference between two discrete variables. It is equal to one when the difference is zero, and zero otherwise.

### Q: How is probability represented for continuous variables?

For continuous variables, probability is represented by a probability density function, which is a continuous function. The probability itself is given by the integral of the probability density over a range of values.

### Q: What is the delta function in the context of continuous variables?

The delta function is the continuous version of the Kronecker delta symbol. It is a function that is zero everywhere except at a specific point, where it is very large. The area under the spike of the delta function is equal to one.

### Q: How can the delta function be thought of intuitively?

The delta function can be thought of as a very high and narrow function with an area of one under the spike. It is a mathematical tool used to study analogs of equations for continuous variables.

### Q: What is the inner product structure for continuous variables?

The inner product between two vectors in the space of functions is given by the integral of the complex conjugate of one function multiplied by the other function.

### Q: What is the inner product between basis vectors of continuous variables?

The inner product between basis vectors is given by the Dirac delta function, which is equal to one when the two vectors are equal and zero otherwise.

### Q: What are the rules of probability in quantum mechanics?

In quantum mechanics, the probability of measuring a particular outcome for a discrete variable is given by the absolute value squared of the probability amplitude. For continuous variables, the probability is given by the square of the wave function amplitude times a probability density.

### Q: What is the condition for periodicity in wave functions for particles on a circular line?

The wave functions for particles on a circular line must satisfy the condition that they come back to the same value after an excursion of the full circumference of the circle.

## Takeaways

The video discussed the concepts of discrete and continuous variables in quantum mechanics and their implications for probability calculations. It highlighted the role of amplitudes and the inner product structure in determining probabilities. The video also introduced the delta function, which is a useful mathematical tool for studying continuous variables. Additionally, it explored the quantization of momentum and angular momentum, showing that they come in discrete values in quantum mechanics. Finally, the video explained how the size of the system affects the spectrum of momentum values, with larger systems having a denser and eventually continuous spectrum.