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I've been trying to get a basic understanding of QM recently. Now, I've stumbled upon this mathematical function called wave function. Here are a few things I know about it.

  • Electrons behave like waves in some situations.
  • A wave function is a complex-valued mathematical function, whose modulus if squared gives the probability density function.
  • The state of a quantum object is completely specified by a wave function.

Now, I'm looking for clarity on two other statements, which I have seen in multiples places on the internet.

  1. A wave function is a mathematical function that shows the "shape" of an electron wave.
  2. A wave function and atomic orbitals are the same thing. A wave function plots the 3D region in the space where you can find an electron.

Are these two statements correct? Or is wave function just a function that is associated with a quantum object from which all the information about its state can be derived? If that's the case, how do we find the wave function of a particular electron? Is it derived from the Schrodinger equation by substituting initial conditions?

Sasikuttan
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  • What do you mean by "electron wave"? – BioPhysicist Oct 01 '21 at 05:28
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    This question is pretty broad. I started making an answer and realized I was going to basically recite a huge section of my Quantum Mechanics book in order to adequately answer it. Also here: "how do we find the wave function of a particular electron?" what context are you interested in exactly? – BioPhysicist Oct 01 '21 at 05:31
  • @BioPhysicist Or rather, the "shape" of an electron. – Sasikuttan Oct 01 '21 at 05:32
  • @BioPhysicist I'm trying to understand it step by step. Electrons behave like waves in some situations, and that's clear to me. No how do we progressively connect this idea with wave function? Just because electrons acts like a wave, we just associate a wave function to it? – Sasikuttan Oct 01 '21 at 05:36
  • @BioPhysicist I don't need any complicated derivations or stuff. I just want to know whether a wave function of an electron is derived from the Schrodinger equation using initial conditions. Just a 'yes' or 'no' will do. I will dig it on my own. Same applies to all other questions, simple one-liners will do. I just need a direction. – Sasikuttan Oct 01 '21 at 05:38
  • maybe this answer of mine will help develop intuition https://physics.stackexchange.com/questions/238855/is-it-wrong-to-say-that-an-electron-can-be-a-wave/238866#238866 – anna v Oct 01 '21 at 07:20

2 Answers2

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Or is wave function just a function that is associated with a quantum object from which all the information about its state can be derived?

The wave function is usually taken to be the state of the particle. In principle, there are other equivalent formalisms which can be used, since they contain exactly the same information (eg. the Wigner quasi-probability distribution). Those can be useful in some contexts, but the wave-function is the more usual formalism.

If that's the case, how do we find the wave function of a particular electron? Is it derived from the Schrodinger equation by substituting initial conditions?

Knowing the wave-function at a time $t_0$, solving the Schrödinger equation will give the wave-function at other times $t$.

A wave function is a mathematical function that shows the "shape" of an electron wave.

Without defining more precisely what you mean by "shape", it is hard to answer this one precisely.

A wave function and atomic orbitals are the same thing. A wave function plots the 3D region in the space where you can find an electron.

In the case of an atom, there are (a discrete infinite number of) stationary states, ie states in which the density probability distribution of an electron will not change in time. Those stationary states are also known as orbitals. Since the wave-function gives you the density probability of the position of the electron, you can compute from it the region where the electron is most likely to be found.

SolubleFish
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  • Thanks, that was helpful. Could you tell me how we could 'know' the wave function at the initial time t0? – Sasikuttan Oct 01 '21 at 06:13
  • @Curiouserandcuriouser usually we can assume that the state is in the ground state. Even if not, we can prepare the system to be in a particular state of our choice by manipulating it. – Superfast Jellyfish Oct 01 '21 at 06:27
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Electrons’ seems like a wave in certain cases and seems like a particle in other. If we want to describe this behaviour mathematically, neither a wave nor a particle is sufficient. Turns out a wavefunction describes this behaviour. A wavefunction describes the state of a system completely (as much is allowed by quantum mechanics).

The mod-square of the wavefunction gives us the probability density function (PDF) of the electron. If you think about it, this is the closest thing to the shape of the electron. What does a PDF represent? It tells us the probability of the location of the electron when it’s position is measured. Now if you image this once, you’ll just get a spot. But if you repeats it many many times, you get the average “shape” of the electron which is just a representation of the PDF.

Atomic orbitals are special wavefunctions that are solutions to the atomic schroginger equation. In other words, orbitals are a subset of possible electronic wavefunctions.

Superfast Jellyfish
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