Lesson: Observables, Expectation Values, and the Measurement Postulate

Introduction:

Welcome to today's lesson on the fundamental concepts of observables, expectation values, and the measurement postulate in quantum physics. These concepts are essential for understanding quantum computing and the behavior of quantum systems.

Observables:

In quantum mechanics, the properties of a system are represented by mathematical operators called observables.
Common observables include position, momentum, energy, and spin.
Each observable has a set of possible values, or eigenvalues, that it can take.

Expectation Values:

The expectation value of an observable is a weighted average of its possible values.
It provides an estimate of the value that the observable is likely to take when measured.
The expectation value of an observable can be calculated using the following formula:

<O> = ∫Psi*(O)Psi dV

where:

Psi is the wavefunction of the system
O is the observable operator
dV is the volume element in which the system is localized

Measurement Postulate:

The measurement postulate states that when an observable is measured, the system collapses into one of its eigenstates with a probability proportional to the square of the modulus of the corresponding eigenfunction.
This means that the act of measurement changes the state of the system from a superposition of states to a single definite state.

Applications in Quantum Computing:

Observables are used to represent the logical gates in quantum computers.
By manipulating the observables, quantum algorithms can perform computations much faster than classical algorithms.

Learning Resources:

Exercises:

Calculate the expectation value of the position operator for a particle in a one-dimensional box.
Explain how the measurement postulate affects the state of a quantum system.
Give an example of how observables are used in quantum computing.