Particle in a box 2

In A Particle in a Box- 1 we have seen that we can write the quantum state of a particle in a box in the eigen-basis of energy operator. From now on we will call this the Hamiltonian operator. For a recap the eigenvalue equation for the Hamiltonian for a particle in a box system is

\hat H \left| \phi_n\right\rangle = E_n \left|\phi_n\right\rangle

with

\left\langle \phi_n|\phi_m\right\rangle=\delta_{nm}

and

\sum_{n=1}^{\infty} \left|\phi_n\right\rangle\left\langle\phi_n\right|=\hat I

With this we could write any arbitrary state of the system as

\left|\psi\right\rangle = \sum_{n=1}^{\infty} \left|\phi_n\right\rangle\left\langle\phi_n|\psi\right\rangle

Here \left\langle\phi_n|\psi\right\rangle are the expansion coefficients (or probability amplitudes) whose modulus square  |\left\langle\phi_n|\psi\right\rangle|^2 is the probability that upon measurement the system will return energy E_n associated with the eigenvector \left|\phi_n\right\rangle.

Lets now ask the question

Where is the particle in the box?

To answer this question we need to look first at Position Operator

 

 

 

 

 

 

 

4 thoughts on “Particle in a box 2”

  1. Pingback: Summary of what we have learned – Quantum & Other Things

  2. These are abysmally amazing articles written in really pellucid and explicit way. Above all they are so easy to understand. They can be made better by adding links for additional reading. Overall they are flabbergasting.

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  3. Sir.
    Can we say that a ket if represented in term of eignvectors(which we know r base vecters) is a superpostion state of a partical .Becz that ket is being represented as liner combination of those eignvectors.

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