# How to find the actual state vector in Quantum Mechanics?

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 ?
• 1. This is like asking, if $F = ma$, then how do you find $F$? It totally depends on the situation you're considering. – knzhou Jun 5 '18 at 16:59
• 2. $|x \rangle$ is defined to be an eigenvector of the position operator. – knzhou Jun 5 '18 at 16:59
• 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?) – Abhikumbale Jun 5 '18 at 17:03
• 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. – knzhou Jun 5 '18 at 17:05
• Could you point to sources or examples where the state vector is calculated? And the coefficients are found ? thanks. – Abhikumbale Jun 5 '18 at 17:05

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 $$|\Psi(t)\rangle = \sum_{\alpha}c_{\alpha}e^{\frac{-iE_{\alpha}t}{\hbar}} |\phi_{\alpha}\rangle.$$