0
$\begingroup$

In most examples I have seen in QM, solutions for bound state of a system is obtained using suitable boundary and normalization conditions. I have not seen examples where the actual state vector or the wave function is calculated and I do not know if one can.

Suppose we label the bound states $\phi_{\alpha}(x)$ where $\alpha$ maybe a continuous label.

Then,

$$\psi (x,t) = \sum_{\alpha}c_{\alpha}e^{\frac{-iE_{\alpha}t}{\hbar}}\phi_{\alpha}(x)$$

Where the summation is replaced by integration if $\alpha$ is a continuous label.

We know that,

$$|\Psi(t)\rangle=\int \psi(x,t) |x\rangle dx$$

I have some questions,

  1. How do you find $c_{\alpha}$ ? Do we need any extra conditions to calculate it?
  2. How do you find $|x\rangle$ the eigenvectors of the position operator?
  3. How does one carry out the integration and find the state vector ?
$\endgroup$
  • 1
    $\begingroup$ 1. This is like asking, if $F = ma$, then how do you find $F$? It totally depends on the situation you're considering. $\endgroup$ – knzhou Jun 5 '18 at 16:59
  • 1
    $\begingroup$ 2. $|x \rangle$ is defined to be an eigenvector of the position operator. $\endgroup$ – knzhou Jun 5 '18 at 16:59
  • $\begingroup$ I know that $|x\rangle$ is defined as an eigenvector of the position operator, but is there any way of finding them out?(Express it in a suitable basis to carry out the integration?) $\endgroup$ – Abhikumbale Jun 5 '18 at 17:03
  • 1
    $\begingroup$ In this context, $|x \rangle$ is your basis. If you want to switch to a new basis, such as perhaps the $|\phi_\alpha \rangle$, then you would need to know $\langle x | \phi_\alpha \rangle$. To do this, you have to solve the time-independent Schrodinger equation. $\endgroup$ – knzhou Jun 5 '18 at 17:05
  • $\begingroup$ Could you point to sources or examples where the state vector is calculated? And the coefficients are found ? thanks. $\endgroup$ – Abhikumbale Jun 5 '18 at 17:05
3
$\begingroup$
  1. How do you find $c_{\alpha}$ ? Do we need any extra conditions to calculate it?

This is like asking "how do I find $\mathbf r(t)$ in newtonian mechanics?", to which the only possible answer is "... for what situation? for which force, and what initial conditions?"

In the case of the dynamics of a quantum particle subject to a time-independent potential $V(x)$, for which you've previously solved the time-independent Schrödinger equation $\left[ - \frac{\hbar^2}{2m}\frac{\mathrm d^2}{\mathrm dx^2} + V(x)\right] \phi_\alpha(x) = E_\alpha \phi_\alpha(x)$ for the eigenstates $\phi_\alpha$, then given an initial condition $\psi_0(x)$ for the system, the expansion you've written, $$ \psi (x,t) = \sum_{\alpha}c_{\alpha}e^{\frac{-iE_{\alpha}t}{\hbar}}\phi_{\alpha}(x), $$ gives the state's evolution starting at $\psi (x,0) = \psi_0(x) = \sum_{\alpha}c_{\alpha}\phi_{\alpha}(x)$ in terms of its basis coefficients in the energy eigenbasis $\phi_\alpha(x)$, which are themselves given by the inner products of the initial condition and the basis functions, $$ c_\alpha = \langle \phi_\alpha | \psi_0 \rangle = \int \phi_\alpha(x)^* \psi_0(x) \mathrm dx. $$ If you don't know the specific initial condition you want to use, then all you've obtained is a general solution that's ready to work when you do, but you cannot use it to say anything concrete about the solution.

  1. How do you find $|x\rangle$ the eigenvectors of the position operator?

You don't "find" them, they're in-built objects of the vector space you started with.

  1. How does one carry out the integration and find the state vector?

You don't. It's a symbolic integration, much like the completely equivalent relation $$ |\Psi(t)\rangle = \sum_{\alpha}c_{\alpha}e^{\frac{-iE_{\alpha}t}{\hbar}} |\phi_{\alpha}\rangle. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.