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Right now I am studying quantum mechanics and I'm having trouble understanding what exactly $\Psi$ is in Schrodinger's equation $\Psi(x) = A\sin(kx) + B\cos(kx)$. After doing some googling I learned that $\Psi$ is the "probability density". Is there any way to take a probability density from a particle in box situation and convert it to a uncertainty that you can use in Heisenberg's Uncertainty Principle?

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closed as unclear what you're asking by Aaron Stevens, Jon Custer, Cosmas Zachos, stafusa, SRS Oct 8 at 9:24

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  • $\begingroup$ If you're studying quantum mechanics, then why do you need to find out what $\psi$ is from googling? Sure it's addressed in whatever source you are studying from? $\endgroup$ – tparker Sep 26 at 3:49
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    $\begingroup$ That’s not the Schrodinger equation. $\Psi$ is the probability amplitude, not the probability density. The probability density is $|\Psi|^2$. $\endgroup$ – G. Smith Sep 26 at 3:50
  • $\begingroup$ The uncertainty principle is a statement about about variances of two hermitian operators so you’d have to figure out first what these are, i.e. what are you computing the uncertainty of? $\endgroup$ – ZeroTheHero Sep 26 at 3:53
  • $\begingroup$ @ZeroTheHero Although the general uncertainty principle also depends on the quantum state as well. $\endgroup$ – Aaron Stevens Sep 26 at 4:04
  • $\begingroup$ Note that the term "wave equation" means something very specific ─ it's restricted to PDEs which govern the evolution of a given system, say, $\frac{\partial^2 f}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}=0$ or, for the Schrödinger equation, $i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi$. The object you've mentioned, $\Psi(x)$, can be termed 'wave function' or 'waveform', or similar terms, but not wave equation. $\endgroup$ – Emilio Pisanty Oct 1 at 12:08
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You are in good company if you are struggling to understand the real significance of the wave function, as many famous physicists have argued about it over the years (see the Wikipedia article on interpretations of quantum mechanics). However, both it and the Heisenberg Uncertainty Principle have well-defined meanings as far as the application of quantum theory is concerned. I will try to explain that here in a simple introductory way.

In quantum theory, the behaviour of a particle is modelled by assuming it has an associated 'wave function', Ψ which varies over time and has some spatial spread. The value of the wave function at any point in space indicates the likelihood of finding the particle there, so a wave-function that is widely spread in space means that the particle's position is quite uncertain.

The theory says that the observable properties of particles are represented by mathematical 'operators'. (If you aren't familiar with that term, look it up, but for now consider it to mean operations, such as d/dx, which convert one function into another.) The possible values of the observable properties of a particle are given by solutions to equations of the form OpΨ=CΨ where Op is the mathematical operator associated with the property, Ψ is the wave function, and C is the value. The Schrodinger equation is an example of such an equation, with H being the operator and E being the allowed values of the energy of a particle.

Typically any of these equations have a range of possible solutions, ie a range of wave functions Ψ and values C that satisfy OpΨ=CΨ. The functions and values that satisfy the equation are called 'eigenfunctions' and 'eigenvalues' of the operator.

Quantum theory says that if you measure a given property of a particle and find it to be a particular eigenvalue of that property, then the wave function of the particle is the corresponding eigenfunction. So if you observe the energy of a particle to be a particular value E, say, then you know that its wave function is given by HΨ = EΨ, which means that it is an eigenfunction of the energy operator H.

The odd aspect of quantum theory is that if you then measure some other property associated with some other operator Op, say, and get a value K, say, for it, then the wave function of the particle has to then be a solution to the equation OpΨ=KΨ, in other words an eigenfunction of Op. Unless Op and H have the same eigenfunctions, that means the wave function of the particle has to 'jump' from being an eigenfunction of H to being an eigenfunction of Op. Weirdly you cannot predetermine which of the eigenfunctions of Op the particle will 'jump' into- all you can do is calculate probabilities for the different options (given by the Born rule).

The Heisenberg Uncertainty Principle relates to this jumping. You can see how it works in the case of energy and position as follows. If you have a particle in a box and it has a specific energy- ie one of the allowed energy eigenvalues- you know that its wave function is the corresponding eigenfunction, which will be spread out in the box. Because it is spread out you cannot be sure where the particle is. If you localise the particle to a region within the box its wave function will be more localised and will no longer be one of the eigenfunctions of energy. If, having localised the particle, you then remeasure its energy, the wave function of the particle will jump unpredictably to become one of the spread-out energy eigenfunctions, so the particle is no longer localised and need not have the same energy eigenvalue it had last time. You simply cannot know the particle's energy and position at the same time as they represent conflicting states of the particle.

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