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I'm trying to solve the Schrödinger equation for a free particle. I've made some progress, but have a few questions that Google hasn't answered.

My first applies to the energy eignenvalue. I don't understand what this is. Is it equivalent to the energy of the particle? Does it apply to free particles? If it does, how is it calculated?

My other question is how to get a wave function from the equation. I don't understand how to get a function by performing operations on scalar values.

I understand the fundamentals of quantum mechanics, but this is my first time doing any calculations. Please take this into account when answering. If you could, please provide a step-by-step answer to guide me through. I really appreciate any help and am totally open to providing clarification.

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  • $\begingroup$ Could you try to add some equations with some illustration of your questions? I don’t understand the getting a wavefunction question particularly well $\endgroup$ – Jake Rose Oct 30 '19 at 21:23
  • $\begingroup$ I think you need to have a proper linear algebra course (this is where you first meet eigenvalues) before delving into QM. From what you say in your question I can tell that you also haven't taken any differential equations course, this is also necessary (Schrödinger's equation is a differential equation after all). If you are bent on studying by yourself I would recommend you find a linear algebra book, see what are the prerequisites and make sure you understand those (if you don't, find books for those subjects and read those first). Then look for a differential equations book and read up... $\endgroup$ – S V Oct 30 '19 at 21:31
  • $\begingroup$ ... to the point where you find higher order differential equations (you may need partial differential equations, but those are for more advanced QM). Afterwards you will need to read something on differential operators (by that time you will know what operators are and should have a good foundation in calculus too), maybe the wikipedia article or something. After that you should be able to solve the Schödinger equation. $\endgroup$ – S V Oct 30 '19 at 21:33
  • $\begingroup$ Thanks for the advice $\endgroup$ – AlexH Oct 30 '19 at 21:35
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    $\begingroup$ @Gert Yes, a free particle is an eigenvalue problem. Eigenvalues can be continuous rather than discrete. $\endgroup$ – G. Smith Oct 31 '19 at 1:48
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We have the Schrödinger Equation for a free particle. $$i\hbar\,\partial_t |\Psi\rangle = \hat{H}|\Psi\rangle $$ The energy eigenstates are the values the particle can take when one physically measures the energy. The probability of each the particle having energy $E$ when measured is given by $\langle\Psi|E\rangle\langle E|\Psi\rangle$. The energy eigenvalues are also the eigenvalues of the Hamiltonian. For the example of the free particle, we are given a Hamiltonian of $\hat{H}=\frac{1}{2m}\hat{P}^2$. Thus: $$\hat{H}|E\rangle = E|E\rangle $$ $$\langle x|\hat{H}|E\rangle = \langle x|E|E\rangle $$ $$-\frac{\hbar^2}{2m}\frac{\partial^2}{{\partial x}^2}\langle x|E\rangle = E\langle x|E\rangle $$ $$\frac{\partial^2}{{\partial x}^2}\langle x|E\rangle = -\frac{2mE}{\hbar^2}\langle x|E\rangle$$ $$\langle x|E\rangle=\alpha\,\mathrm{exp}\left(i\frac{\sqrt{2mE}}{\hbar}x\right)+\beta\,\mathrm{exp}\left(-i\frac{\sqrt{2mE}}{\hbar}x\right)$$ Once we have an energy wavefunction, we must solve for the actual wavefunction. I'll go through the derivation of the propagator for you, so you understand the general process. $$i\hbar\,\partial_t |\Psi\rangle = \hat{H}|\Psi\rangle $$ $$\langle E | i\hbar\,\partial_t |\Psi\rangle = \langle E |\hat{H}|\Psi\rangle $$ $$i\hbar\,\partial_t \langle E |\Psi\rangle = E\langle E |\Psi\rangle $$ $$\partial_t \langle E |\Psi\rangle = -\frac{iE}{\hbar}\langle E |\Psi\rangle $$ $$\langle E |\Psi\rangle= A \mathrm{exp}\left(-\frac{iE}{\hbar}t\right)$$ $$\langle E |\Psi\rangle= \mathrm{exp}\left(-\frac{iE}{\hbar}t\right)\langle E|\Psi(0)\rangle$$ Now, we have \begin{align} |\Psi\rangle & = \int_{-\infty}^{\infty}|E\rangle\langle E |\Psi\rangle dE\\ & = \int_{-\infty}^{\infty}|E\rangle\langle E|\Psi(0)\rangle\,\mathrm{exp}\left(-\frac{iE}{\hbar}t\right) dE \end{align} \begin{align} \langle x|\Psi\rangle & = \int_{-\infty}^{\infty}\langle x|E\rangle\langle E|\Psi(0)\rangle\,\mathrm{exp}\left(-\frac{iE}{\hbar}t\right) dE \\ & = \int_{-\infty}^{\infty}\langle x|E\rangle\int_{-\infty}^{\infty}\langle E|x\rangle\langle x |\Psi(0)\rangle dx\,\mathrm{exp}\left(-\frac{iE}{\hbar}t\right) dE \end{align} Now, given an inital wavefunction, you can calculate the wavefunction in position space.

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