Difference bettwen a wavefunction and state vector (with an specific example)

Question

I am trying to get my head around the difference between wavefunction and statevector. I looked at the previous answers in this site and don't full understand. Can you some explain the following.

Example: Consider the particle in a infinite potentional:

Here the wavefunction in terms of energy is $ψ_{n}=\sqrt{\frac{2}{L}}\sin(\frac{nπ}{L}x)$.

A state vector is terms of energy eigenfunction is $|ψ⟩=\sum_nc_i|i⟩$

• Here $|i⟩$ is the energy eigenfunction given by $ψ_{n}=\sqrt{\frac{2}{L}}\sin(\frac{nπ}{L}x)$
• $c_i$ is some constant

A state vector written in terms of wave function: $|\psi\rangle = \int d^3r\;\psi(\mathbf{r})|\mathbf{r}\rangle$ (taken form State Vector vs wave function)

• Consider the case where the particle is not in superpostion but in one state {n}.
• Here |r⟩=|i⟩= $ψ_{n}=\sqrt{\frac{2}{L}}\sin(\frac{nπ}{L}x)$
• Thus: $|ψ⟩=\int \psi_{n} \psi_{n} d^3r$ (???)

Answer 1

In some sense this looks like you kind of have the right idea... I might add the caveat "if that's how you want to think about it". One thing that's definitely wrong is you're messing up index notation in a few places, and I think it's likely that if you fix that things will make a lot more sense to you.

Make sure any index that shows up in your equation is either defined by the summation variable or appears on the left hand side of the equation!! $$\sum_n c_i |i\rangle\text{ }\color{red}\times$$ dosnt make sense... $i$ is not defined here. What you mean is $$\sum_i c_i |i\rangle\text{ }\color{green}\checkmark$$ Then later you have $$|\psi\rangle=\int \psi \psi_n dx\text{ }\color{red}\times$$ It should really set you off that the left side is a state vector and the right side is just a number. What you mean is $$|\psi\rangle=\sum_i\int \psi \psi_i dx |\psi_i\rangle\text{ }\color{green}\checkmark$$ So that we get $$ c_i=\int \psi \psi_i \text{ }\color{green}\checkmark $$ And finally you make another kind of error - defining $n$ in two different ways. $n$ can't be both the summation variable and a variable on the left hand side of the equation. So if you want to define $n$ as the wave function in question, then the summation variable needs to be different, say $i$. $$ |\psi_n\rangle=\int\psi_n\psi_n d^3r\text{ }\color{red}\times $$ $$|\psi_n\rangle=\sum_i\int \psi_n \psi_i dx |\psi_i\rangle=\sum_i\delta_{ni} |\psi_i\rangle=|\psi_n\rangle\text{ }\color{green}\checkmark$$ Where I have used the orthonormality of $\psi_i$ to complete the integral and then the sum.