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

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: $$|ψ⟩=\int \psi_{n} \psi_{n} d^3r$$ (???)
• Where did you get $|\psi \rangle = \int \psi_n \psi_n dx$ ? Commented Jul 14, 2023 at 12:45
• @Tensor I added more information. What would the correct equation for |ψ⟩ be Commented Jul 14, 2023 at 12:53
• As a simpler example - a vector can be represented abstractly as $\vec{v}$ or as a set of 3 numbers $(v_x,v_y,v_z)$. The latter representation describes the components of $\vec{v}$ in the $(\vec{x},\vec{y},\vec{z})$ basis. If this makes sense to you, then using the same language -- "A state can be represented abstractly as $| \psi \rangle$ or as $\psi(x)$. The latter representation describes the components of $|\psi \rangle$ in the $|x\rangle$ basis." Commented Jul 14, 2023 at 13:02
• I'm not an expert on this. But as far as I know, state vectors are equivalent to wavefunctions, but written in different notation. Most of the confusion usually come from the notations. I think this link give a pretty decent introduction. For quick test, consider ground state solution to Schrodinger equation, and how will you include spin into the solution. Commented Jul 14, 2023 at 13:03

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.
• So in the example i have given are $|\psi_{n}⟩$ and $\psi_n$ the same. With them being $ψ_{n}=\sqrt{\frac{2}{L}}\sin(\frac{nπ}{L}x)$. Also where did you get $$|\psi\rangle=\sum_i\int \psi \psi_i dx |\psi_i\rangle\text{ }\color{green}\checkmark$$from, the equation i found in the link was different. Commented Jul 14, 2023 at 13:07
• @PhysicsQuestion In the linked question, they focus particularly on converting a state vector into a wavefunction by applying all the same equations I have provided in the position basis. So instead of having discrete different vectors (in an arbitrary basis) $|\psi_n\rangle$ and discrete wavefunctions $\psi_n$, they have a continuous basis of position-space eigenfunctions $|\psi_x\rangle$, where $\psi_x(x')=\delta(x'-x)$. So maybe my equations look different, but they're actually the same equations, and the people in the link have inserted a particular set of "basis" wavefunctions. Commented Jul 14, 2023 at 13:17
• @PhysicsQuestion I'd say the two are the "same thing" in some sense... they represent the same physical state for the particle. But in another sense they are different - one is a special notation that is a faster way of writing the wavefunction in the position-basis. When you write $\psi_n=$ I expect you to write a function of $x$ on the other side. Might be nice to write it as $\psi_n(x)$. An explicit equation that relates the two is $\psi_n(x)=\langle x|\psi_n\rangle=\int \psi_n(x')\psi_{x} dx'$ $=\int \psi_n(x')\delta(x-x') dx'=\psi_n(x)$ Commented Jul 14, 2023 at 13:21