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this might be a question that does not have an answer. I am working out the formula for energies for the anti-symmetric wave functions for the infinite square well. I need to determine the value for $\xi_0$ which yields a bound state. I am struggling to understand what $\xi_0$ is.

From notes and asking my teacher, I am told that $\xi = a\sqrt{\frac{2mE}{\hbar^2}}$; where $m$ is mass, $E$ is the total energy of the particle, $ \hbar$ is planks constant, and $a$ is the length or width of the square well.

Question: What does $\xi$ represent?

Context to $\xi$: It seems like an arbitrary value. Although, from the context of the square well problem I can find something similar. Take the time independent scrodinger equation and set $U(x)=0$, because it has no potential energy while inside the walls of the well. $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + U(x)\psi(x)= E\frac{d\psi(x)}{dt}\\ [U(x) = 0]\\ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} = E\frac{d\psi(x)}{dt}$$ Then divide both sides by $-\frac{\hbar^2}{2m}$. $$\frac{d^2\psi(x)}{2m}=-\frac{2mE}{\hbar^2}\frac{d\psi(x)}{dt}$$ For convenience, make the definition $k^2\equiv \frac{2mE}{\hbar^2}$. Then $ak = \xi$.

The anti-summetric solution looks like the following, inside the well $$\psi_n(x)=\sqrt{\frac{2}{L}}sin(\frac{n\pi x}{a});\qquad0<x<a$$ $n$ is an integer. The quantized energy states that form because at $\psi(0)=\psi(a)=0$ and $\psi$ has to be normalized produces $$E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}$$

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The de Broglie wavelength of a free quantum particle is $$\lambda = \frac{h}{p} = \frac{h}{\sqrt{2mE}}.$$ Therefore, up to a constant factor, we just have $$\xi = \frac{a}{\lambda}.$$ That is, $\xi$ is a dimensionless measure of how wide the well is, compared to the de Broglie wavelength of the particle.

It's useful for work with $\xi$ instead of $a$ because it lets us "factor out" a bunch of constants we don't want to carry around, and the final answer will be something nice like "$\xi = 1$" instead of something messy like "$a = 4.7239 \times 10^{-14} \text{ meters}$". Phrasing the answer solely in terms of $\xi$ will also automatically tell you how the value of $a$ should change as other parameters, like $m$ and $E$, change as well.

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  • $\begingroup$ So if $a,m$ is fixed/given, then $\xi$ turns into a dimensionless measure of the energy of a particle compared to the size of the well? $\endgroup$
    – Tsangares
    Oct 15, 2016 at 20:53
  • $\begingroup$ @Tsangares Sort of. It's a bit awkward to say that because it's proportional to $\sqrt{E}$, not $E$, but yes. It measures how big the energy is, by comparing the resulting de Broglie wavelength to the length scale of the well. $\endgroup$
    – knzhou
    Oct 15, 2016 at 20:55
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Your given sine wave eigenfunctions are anti-symmetric in the well only for $n$ is even. The parameter $\xi = a\sqrt{\frac{2mE}{\hbar^2}}=ka$ is equal to the product of wave vector k and the width of the well a which is the dimensionless argument of the sine wave function.The boundary conditions of the well restrict the values of $\xi$ to $\xi_n =ka=n\pi x$ with the symmetric solutions for $n=1,3,5,..$ and the anti-symmetric solutions for $n=2,4,6,...$. All these sine eigenfunctions are bound states.

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