# Eigenstates of the Hamiltonian | Summary and Q&A

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July 5, 2017
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MIT OpenCourseWare
Eigenstates of the Hamiltonian

## TL;DR

This content introduces the concept of spin, eigenstates, and spin directions, highlighting their importance in quantum systems and practical applications like quantum computing.

## Key Insights

• ❎ Angular momentum and spin can be measured using operators like L squared, Lz, spin squared, and Sz.
• ❓ Eigenstates, which have definite values for an observable, can be found for spin by determining their eigenvalues.
• ☺️ By forming linear combinations of spin up and spin down states, spin states can be obtained in different directions like x and y.

## Transcript

PROFESSOR: I want to just elucidate a little more what are the eigenstates here. So with angular momentum, we measure L squared and we measure Lz. So with spin, we'll measure spin squared and Sz. And Sz is interesting. It would be spin or angular momentum in the z direction. So let's look at that, Sz, this is the operator, the measurable. It's this... Read More

### Q: What are eigenstates in the context of angular momentum and spin?

Eigenstates are states in which a given observable, like angular momentum or spin, has a definite value. In the case of spin, eigenstates are determined by their eigenvalues, which represent the component of spin in a specific direction.

### Q: How are spin up and spin down states related to eigenstates?

Spin up and spin down states are eigenstates of the operator Sz, representing the z component of spin. Spin up corresponds to a positive eigenvalue (h bar over 2), while spin down corresponds to a negative eigenvalue (-h bar over 2).

### Q: Can spin states exist in directions other than z?

Yes, by forming linear combinations of spin up and spin down states, spin states can be obtained in different directions. For example, a linear combination of spin up and spin down states can result in a spin state that points up in the x direction.

### Q: How does the use of complex numbers play a role in obtaining spin states?

Complex numbers are crucial in forming linear combinations and obtaining spin states in different directions. The presence of complex numbers allows for the superposition of spin up and spin down states, leading to spin states pointing in all possible directions.

## Summary & Key Takeaways

• The professor explains the concept of eigenstates in angular momentum and spin and how they are measured using L squared, Lz, spin squared, and Sz.

• Eigenstates for spin are easily found, with spin up and spin down states being determined by their eigenvalues.

• The content explores how to obtain spin states in different directions, such as x and y, by forming linear combinations of spin up and spin down states.