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In quantum mechanics, in the 'particle in a box' topic, I studied that by solving the Schrödinger equation, we can actually find out what the wave function of the particle in a box looks like. Corresponding to different energy levels, the particle can have different wave functions. For e.g., in it's ground state (lowest energy level), the wave function looks like a standing wave (with nodes at the edges of the box) if we just look at the real part of the wave function. But in reality if we look at the entire wave function (including the imaginary part), it's wave function is actually a wave that is rotating through the real and the imaginary planes. There are plenty of animations in YouTube regarding this.

Similarly, what does the wave function of an electron look like? In my book, no description has been given as to what the wave function of an electron looks like. I mean what does the plot of $Ψ(r,θ,\phi,t)$ (since we use spherical polar coordinates while solving the Schrödinger equation for an atom; $t$ is time) look like?

I don't want to just see the time-independent components of the wave function. I want to see what the wavefunction look like if we let time run (for e.g., for a particle in a box, the wave function just looks like a static wave but when we let time run, the wave function is actually a wave rotating through the real and imaginary planes). Can someone please show me a plot or animation of $Ψ(r,θ,\phi,t)$ so that I can visualise what the wave function of an electron looks like? For instance, what does the wave function of the electron in, say, a hydrogen $1s$ orbital look like? Similarly, what does the wave function of electrons in other orbitals look like?

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  • $\begingroup$ The wave function of the 1s electron if we let time run looks exactly like the wave function of the 1s electron. It is stationary. It is the very thing you don't want to see. What is it that you want to see, then? $\endgroup$ May 27 at 18:31
  • $\begingroup$ There was actually a recent MinutePhysics video that attempted to provide a better visual of orbitals in motion. $\endgroup$
    – Tyberius
    May 27 at 19:15
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    $\begingroup$ $2p_x$ does not evolve with time. $\endgroup$ May 28 at 5:34
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    $\begingroup$ Do you mean an electron in an atom, in which case the wavefunction is that of an electron in a potential ,say 1S, 2P etc., or a free electron? $\endgroup$
    – porphyrin
    May 28 at 6:38
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    $\begingroup$ I think that this question touches elements not properly covered in basic and even specialised courses and does not deserve downvotes unless OP is a somewhat graduated or a teacher. Opposite it lead to nice and useful comments and answer. $\endgroup$
    – Alchimista
    May 28 at 11:30
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Stationary wavefunctions are solutions of the time-dependent Schrödinger equation $$\hat{H}\psi=i\hbar\frac{\partial \psi}{\partial t} $$

for which the energy $E$ is constant (it being - like the Hamiltonian - otherwise generally time-dependent) so that $$\begin{align}\hat{H}\psi&=E\psi\\&=i\hbar\frac{\partial \psi}{\partial t}\end{align} $$ which has solutions $\psi(t)=\psi(0)\exp(-i\phi(t))$. Here $\phi(t)=\omega t$ is a time-dependent phase factor, and $E=\hbar \omega $ is the familiar Planck-Einstein relation between frequency and energy.

The phase $\phi(t)$ is a funny thing because, although you can relate the relative phase of one wavefunction at two points in time or of the wavefunctions of two particles, you can't afaik picture the phase in some way that makes sense in a macroscopic (classical physics) sense. In other words, the phase might change, but what that is that is changing is not something that can be explained outside of mathematics. There is no coordinate system that will give meaning to the QM phase (unlike the amplitude of a classical wave, which can in general be related to spatial coordinates, or some physically measurable amplitude).

If you want to visualize then how the wavefunction (or its phase) evolves in time, in particular for a $\mathrm{1s}$ orbital, imagine a sphere representing an isosurface of the radial component of the wavefunction, where the surface color oscillates between values representing different phases (from 0 to 360 degrees), something like this:

enter image description here

(red, yellow, green and blue represent sequential 90 degree phase rotations)

The time-dependent phase of any orbital representing a stationary solution of the Schrödinger equation can be similarly represented as a regular oscillation between a series of colors, with regions of the orbital with the same phase being uniformly colored.

Note that the spatial contribution to a wavefunction can also be complex valued, so that different parts of an orbital can differ in both phase and amplitude. For instance, the lobes in a p-orbital represent a surface of uniform time-independent amplitude, but the phase changes sign across a node. Neither the spatial phase factor nor the time-dependent phase factor alter the amplitude of the wavefunction.

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    $\begingroup$ It might have been easier to show the behavior of a 2 p orbital with its famous dumb-bell shape. Here the first half-dumb-bell is green when the second one is red. And after a while, without moving, the first half-dumb-bell turns red when the second turns green, and so on. $\endgroup$
    – Maurice
    May 27 at 19:15
  • $\begingroup$ @Buck Thorn, can you give the animations of wave functions of electrons in other orbitals also? $\endgroup$ May 28 at 4:38
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    $\begingroup$ You might says that your animation is slowed down. Well done. $\endgroup$
    – Alchimista
    May 28 at 13:16
  • $\begingroup$ @Alchimista Yes, maybe that's not the 1s orbital, based on the frequency you can estimate what is the principal Q# :) $\endgroup$
    – Buck Thorn
    May 28 at 17:05
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    $\begingroup$ @Alchimista thanks for your comment. Yes, this is tricky because I am showing only how the time-dependent part of the wavefunction evolves, and of course the phase is a product of time-dependent and independent parts. I'll add that to my answer when I get a chance. Plus I realized that I should have used two spheres, not one, so I need to edit anyway. $\endgroup$
    – Buck Thorn
    May 29 at 9:33

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