Completeness of eigenvectors and measurement postulate

TL;DR

Any physical state can be written as a superposition of eigenfunctions, with probabilities given by the square of the expansion coefficients.

Transcript

PROFESSOR: That brings us to claim number four, which is perhaps the most important one. I may have said it already.b The eigenfunctions of Q form a set of basis functions, and then any reasonable psi can be written as a superposition of Q eigenfunctions. OK, so let's just make sense of this. Because not only, I think we understand what this means,... Read More

Key Insights

• 😫 Eigenfunctions of a Hermitian operator form a complete set of basis functions for representing physical states.
• ❓ Expansion coefficients determine the probabilities of obtaining specific measurement outcomes.
• 👋 The calculation of expansion coefficients involves taking the integral of the product of eigenfunctions and the wave function.
• 👋 Normalized wave functions have squared expansion coefficients that sum up to 1, ensuring the conservation of probability.
• 🦾 Measurement in quantum mechanics involves obtaining one of the eigenvalues of a Hermitian operator.
• 👻 The measurement postulate allows the prediction of measurement outcomes and the collapse of the wave function after measurement.
• ❓ Different Hermitian operators correspond to different measurements, and their eigenfunctions are used to represent the physical states.

Questions & Answers

Q: What does it mean to write a physical state as a superposition of eigenfunctions?

Writing a physical state as a superposition means expressing it as a linear combination of eigenfunctions, with expansion coefficients determining the probabilities of obtaining specific measurement outcomes.

Q: How are the expansion coefficients alpha i calculated?

The expansion coefficients can be calculated by taking the integral of the product of the eigenfunction and the wave function. This integral represents the overlap between the two functions and gives the value of alpha i.

Q: What is the significance of the sum over i of alpha i squared equaling 1?

If a wave function is normalized, then the sum of the squared expansion coefficients must equal 1. This ensures that the probabilities of obtaining any measurement outcome add up to 1, guaranteeing the conservation of probability.

Q: What happens to the wave function after a measurement?

After a measurement is made and a specific measurement outcome is obtained, the wave function "collapses" to the corresponding eigenfunction associated with that outcome. This is called the collapse of the wave function.

Summary & Key Takeaways

• The eigenfunctions of a Hermitian operator form a set of basis functions that can be used to represent any physical state.

• The expansion coefficients, alpha i, determine the probabilities of obtaining a specific measurement outcome.

• The calculation of alpha i is straightforward by taking the integral of the product of the eigenfunction and the wave function.