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I know that the square of the wave function can tell me the most probable location of the particle, but let's say I didn't square the function and I'm only looking at a graph of the wave function itself. What should I be able to tell from its graph?

My question was inspired by these graphs from Introduction to Quantum Mechanics by Griffiths:

enter image description here

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    $\begingroup$ How are you going to graph the complex number? $\endgroup$
    – zeldredge
    Oct 6 '16 at 23:41
  • $\begingroup$ @zeldredge see edited answer. I can kinda see what you're implying but then how do you explain these graphs by D.J. Griffiths? $\endgroup$ Oct 6 '16 at 23:45
  • $\begingroup$ How wiggly it is tells you how kinetic-energy-y it is, for instance. In technical terms, the kinetic energy is the con abut of the graph, and faster wiggles generally implies larger concavities. $\endgroup$
    – march
    Oct 7 '16 at 1:17
  • $\begingroup$ @march what is "con abut"? $\endgroup$ Oct 9 '16 at 19:29
  • $\begingroup$ That was supposed to say "concavity". Autocorrect. $\endgroup$
    – march
    Oct 9 '16 at 20:11
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I know that the square of the wave function can tell me the most probable location of the particle

The term 'location' is somewhat problematic here.

For particle in a 1D well with a normalised wave function $\psi$, the Born interpretation tells us:

$$P[x_1,x_2]=\int_{x_1}^{x_2}\psi^*\psi dx$$

Or if $\psi$ is Real, then $\psi^*=\psi$, so:

$$P[x_1,x_2]=\int_{x_1}^{x_2}\psi^2 dx$$

Schematically:

Born interpretation

This gives us the probability of finding the particle on the interval $[x_1,x_2]$. Note that when the interval goes to $0$ (because $x_1=x_2$) then $P=0$.

The probability of finding the particle in a point location is therefore always zero and this is in agreement with Heisenberg's Uncertainty Principle. $\psi^*\psi$ is therefore better referred to as the probability density function, rather than the misleading probability. We can only calculate a probability over a specific area of space, not in a point location.

The wave function $\psi$ shows that it changes signs at the nodes (roots, i.e. zero points). This finds important applications in chemistry because atomic wave functions (atomic orbitals) can only interact additively, thereby forming molecular wave functions (molecular orbitals or bonds), if they are of the same sign.

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  • $\begingroup$ The probability of finding the particle in a point location is therefore always zero - am I correct in thinking that it actually makes sense intuitively that we can't find a particle in a point location because if we imagine a physical wave as the correct representation of the particle, then simply pointing to one point on the wave is meaningless - we must talk about the entire wave in order to talk about where it is. $\endgroup$ Oct 9 '16 at 19:52
  • $\begingroup$ Yes, that's more or less correct. More accurately is to invoke the Uncertainty Principle: we cannot know position and momentum both with arbitrary precision: $\sigma_x \times \sigma_p \geq \hbar/2$. Set $\sigma_x=0$ and the principle is violated. $\endgroup$
    – Gert
    Oct 9 '16 at 20:53

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