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how do you find the expectation value <x> for the 1st 2 states of a harmonic oscillator?
Galileo said:Well, they are [itex]\langle \psi_0|x|\psi_0\rangle[/itex] and [itex]\langle \psi_1|x|\psi_1\rangle[/itex] ofcourse.
You could find them either by integration or the application of the ladder operators.
However, a look at the probability distributions [itex]|\psi_0|^2[/itex] and [itex]|\psi_1|^2[/itex] should tell you immediately what the expectation value for the position is.
A harmonic oscillator is a system in which the restoring force is directly proportional to the displacement from equilibrium. This results in a sinusoidal motion. Examples of harmonic oscillators include a mass attached to a spring and a pendulum.
An expectation value is the average value of a measurable quantity in a given system. In the context of quantum mechanics, it is the average value of an observable, such as position or momentum, for a particular state of a system.
To find the expectation value for the first two states of a harmonic oscillator, you need to calculate the integral of the wavefunction squared multiplied by the corresponding observable. For the position operator, the integral is ∫ψ* x ψ dx, and for the momentum operator, it is ∫ψ* x ψ dx.
The wavefunction of a harmonic oscillator is a mathematical function that describes the probability of finding a particle in a particular state, such as position or momentum, in a harmonic oscillator system. It is represented by the Greek letter psi (ψ) and can be calculated using the Schrödinger equation.
Finding the expectation value for the first two states of a harmonic oscillator is important because it allows us to understand the average behavior of a particle in a harmonic oscillator system. This information can be used to make predictions about the behavior of the system and to compare it to experimental results. It also provides a way to quantify the uncertainty in the position and momentum of the particle.