How to Represent a Wavefunction in Dirac Notation for an Infinite Square Well?

[v_pino]

Homework Statement

For the infinite square well, a particle is in a state given by $\psi = \frac{1}{\sqrt 2}(\psi_1 + \psi_3)$ , where $\psi_1$ and $\psi_3$ are energy eigenstates (ground state and the second excited state, respectively).

Represent this state as a column matrix $\psi>$ in the energy basis and x basis. You may use your knowledge of the solutions of the infinite square well from before, obtained in the x basis. State with the help of mathematical equations how you would find the column matrix in k basis.

Homework Equations

I know that in the x-space, the column matrix representation of basis vector is |x> and the components of a state vector $\psi$ is $<x|\psi>$. And likewise, replace 'x' with 'k' for the k-space basis.

The Attempt at a Solution

Is writing $\psi = \frac{1}{\sqrt 2}(\psi_1 + \psi_3)= \frac{1}{\sqrt 2}(<i|\psi_1>+<i|\psi_3>)$ permitted? If not, can you please point me in the right directions? I have Gritffiths Introduction to QM book so any reference to that I can get hold of. Thanks.

 

[mark726]

Thank you for your question. To represent the state \psi as a column matrix in the energy basis, we can use the energy eigenstates \psi_1 and \psi_3 as our basis vectors. This means that our column matrix \psi> will have two components, one for \psi_1 and one for \psi_3 . We can write this as:

\psi> = \begin{bmatrix} <\psi_1|\psi> \\ <\psi_3|\psi> \end{bmatrix}

Using the knowledge that <x|\psi> represents the component of a state vector in the x basis, we can write this as:

\psi> = \begin{bmatrix} <x|\psi_1> \\ <x|\psi_3> \end{bmatrix}

To find the column matrix representation in the k basis, we can use the Fourier transform relation between the x and k basis vectors, which is given by:

<x|k> = \frac{1}{\sqrt{2\pi}} e^{ikx}

Therefore, we can write the column matrix representation in the k basis as:

\psi> = \begin{bmatrix} <k|\psi_1> \\ <k|\psi_3> \end{bmatrix}

To find the components of this column matrix, we can use the knowledge that the energy eigenstates are given by:

\psi_n(x) = \sqrt{\frac{2}{L}} sin \left(\frac{n\pi x}{L}\right)

Where L is the length of the infinite square well. Therefore, we can write the components of the column matrix as:

<k|\psi_n> = \sqrt{\frac{2}{L}} \int_{-\infty}^{\infty} sin \left(\frac{n\pi x}{L}\right) e^{-ikx} dx

Solving this integral will give us the components of the column matrix in the k basis. I hope this helps. Let me know if you have any further questions.