# What precisely is the wave function a probability density of?

In QM the norm of the wave function $\psi(\vec{x})$ is said to be the probability density that the particle is at $\psi(\vec{x})$ if one would observe its position. Generally, nothing more is explained what this exactly means, yet I'm confused by it.

Hence my question: Does it mean that any given time, say, the electron in ground state of Hydrogen is at some exact position $\vec{x}$, but we just do not know exactly where, unless we measure it?

If so, are such measurements actually done, e.g. was the location of the electron found at location $\vec{x}$, say, so many Angstrom from the nucleus at these polar angles in a particular measurement (given if repeated many times it should follow the distribution given by the wave function).

Note: The interpretation the electron is extended in space while not intuitive would not be that confusion to me but yet I always see it described as a probability density of being located at some $\vec{x}$ which implies it can be observed to be at some exact location.

• Anytime someone says an electron has a given probability to be some, they're using sloppy language. The correct phrasing is to say that you have a certain probability of finding it as some position when you measure, which is not the same thing. The position of the electron simply doesn't have a well defined value before you measure it. Apr 18, 2017 at 18:43
• > "In QM the norm of the wave function ψ(x⃗ ) is said to be the probability density that the particle is at ψ(x⃗ ) if one would observe its position." I have never hear or read that in my studies. Did you mean that $|\psi(x)|^2$ is value of probability density of particle position, at point $x$? This is what Born's rule (one of two) states. Apr 18, 2017 at 21:37
• @Jan Lalinsky Indeed - sloppy language on my side - will correct. Apr 19, 2017 at 3:00
• @Javier I understand the point that it has no defined value unless you measure it. Yet I have not seen data (before my question) on experiments that found it somewhere when doing measurements similar to e.g. the speckles observed in a low intensity double slit experiment with light. Conifold referred to another question that addressed this with references. Apr 19, 2017 at 3:24
• I'm with @JánLalinský on this one. $\psi$ is not a probability density and it is never referred to as such. You can call it a probability (density) amplitude if you want, but that's a very different beast. This post does need to be edited. Apr 20, 2017 at 18:16

To be precise, it is not the complex-valued wave function $\psi(\vec{x})$ that is interpreted as a probability density, but its absolute square $|\psi(\vec{x})|^2$ is, sometimes $\psi(\vec{x})$ is called probability wave to stress the difference. This is significant, if $\psi(\vec{x})$ itself was a density, or if $|\psi(\vec{x})|^2$ itself obeyed an evolution equation, classical statistical interpretation of them as particle densities might have been possible, as in the Fokker–Planck equation. As it is, the probabilistic interpretation does not mean that electron is at some exact but unknown position at some time $t$, it only means that if a measurement is performed at that time, then the probability of detecting it in some region is equal to the integral of $|\psi(\vec{x})|^2$ over that region, see the Born rule.

The "measurement" roughly means that the electron is made to interact with another system, which transforms its wave function into an approximation of one of the eigenfunctions of the position operator (which are $\delta$ functions). It makes no sense to ask where electron "is" while it is in a superposition state $\psi(\vec{x})$ (it is a superposition of it being everywhere), only for position eigenstates the question is meaningful. So it makes no sense to ask where the electron is unless the measurement is actually performed (or the state happens to be localized for some other reason). Only half of the "measurement" process is non-controversially understood, see decoherence. The other half, namely how the eigenfunction in question is selected, is answered differently by different interpretations of quantum mechanics. In the Copenhagen interpretation, for example, the measurement indeterministically "collapses" the wave function into one of the eigenfunctions, and integral of $|\psi(\vec{x})|^2$ gives the probability of the collapsed eigenfunctions being localized in a particular region.

To determine the position of an electron it must be probed, the more precision is sought the stronger the probing interaction must be. One can bombard an atom with particles to knock an electron off of its orbit, then detect the scattered electron and infer its position by backtracking the evolution. Something like this is done in photoionization microscopy, which was used by Stodolna et al. in 2013 to image the hydrogen atom's wave function (but they did not do it by knocking off one electron at a time). Or one can scatter the particles off of the atom and detect them, as Rutherford did. Neither is sensitive enough to pinpoint individual electron locations within an atom, usually only statistical data is collected. Even if they did the atom would be destroyed as a result of such "pinpointing" due to the required interaction energies, so the measured location will no longer be "within the atom" after the collapse. So-called nondemolition measurements, which preserve the atom, would not be able to do the pinpointing, see What is the experiment used to actually observe the position of the electron in the H atom?

• Your first sentence captures my confusion. It has no exact position at a given time yet you can determine it was in a certain region (which I assume could be small with respect to the atom size) after detecting it. This sounds like a contradiction. The second part of my question was about actual measurements of it being in a certain region. I only see these extended blobs of e.g. the 1s wave function. PS: by an exact location I mean some region in space which is small in comparison with the atomic radius. Apr 18, 2017 at 3:17
• @JanBos Your confusion comes from thinking of electron classically as some pre-existent "point" or "ball". There is none of that. It does not exist as such until a measurement is made, it exists only as a superposition of all states. The localized "thing", which is detected, is only generated by the process of measurement itself (because of the collapse). So what you "determine" is not what was in the region, but what it became after the measurement. Apr 18, 2017 at 4:04
• The whole picture of electrons having positions (to those precisions) while we are not looking is misleading, to answer your second question. And with the position measured to such a precision the uncertainty in the momentum will be so huge that the next instant that electron may be detected light years away from that atom. Performing a measurement in this situation would defeat its purpose. It's not that electrons do not exist when we are not looking, but their existence is not imaginable in any familiar terms, like waves or particles. Apr 18, 2017 at 4:16
• No I'm not thinking of electrons as points or balls. See the 2nd paragraph part of my question in which the extended nature of electrons can make sense. My problem is that by authors calling the wave a "probability" to be located in a region drives one to think of them as points/balls which leads to the contradiction. Apr 18, 2017 at 5:47
• @JanBos Keep in mind that the speckles on the screen or tracks in the Wilson chamber are highly misleading as "locations" (they are outcomes of cascading processes only triggered by particles). The fact that they are visible with a naked eye already shows that they are many times larger than what is being "located". "Speckles" at subatomic scales would require far more energetic measurements, so once an electron is localized by collapse the atom is no more. This is how the uncertainty principle makes itself felt even when you are not attempting to measure position and momentum simultaneously. Apr 19, 2017 at 2:08

The electron is an elementary particle in the standard model of particle physics. As such it is postulated to be a point particle. BUT this is quantum mechanics, the term particle is just a tag on a quantum mechanical entity.

Quantum mechanical entities follow boundary conditions in completely determined solutions of quantum mechanical equations. Within these solutions they are described by a function F(x,y,z,t) but this is a complex function and can acquire a physical meaning within an expectation value of a position operator or a momentum operator, which involves the complex conjugate squared,. It is a postulate of quantum mechanics that this is a probability density for finding the electron. One has to do many measurements with the same boundary conditions to get the probability distribution.

In the case of the double slit single electron at a time one can see the probability density developing because the electrons are not in a bound state. A single electron interacts with the screen and leaves a point characteristic of a classical particle. It is not spread out over the whole pattern.

Electrons in atoms are described by the orbitals and as Conifold has referred also the Hydrogen orbitals have been measured in a specific experiment: It is a probability density distribution, a collective measurement of interactions with photons

After zapping the atom with laser pulses, ionized electrons escaped and followed a particular trajectory to a 2D detector (a dual microchannel plate [MCP] detector placed perpendicular to the field itself). There are many trajectories that can be taken by the electrons to reach the same point on the detector, thus providing the researchers with a set of interference patterns — patterns that reflected the nodal structure of the wave function.

The x and y coordinates are in mm on the detector plane.

The color coding is the intensity registered at the detector. A point on the plot builds the interference patterns of the position of the electrons. In a sense , the double slit interference pattern reflects the geometry of the two slits and one could arrive at the geometry by using the pattern. The interference pattern seen above can be correlated with the calculations of the orbitals. For details the paper can be seen here.

From this data an estimate within the Heisenberg uncertainty could be made of the position of the probable d(V) of the electron in the atomic dimensions but as the hydrogen atom has a complete quantum mechanical solution, one uses the mathematics of orbitals.

• Thanks! It is not clear to me what the x and y coordinates in this graph stand for. The location on the detector the zapped electrons arrived? Secondly, what does the color represents? Thirdly, when e.g. backtracking is there some correlation with the region around the nuclear the interaction took place. I will try to read the papers but it will be good to add that to your answer. Apr 20, 2017 at 3:55
• It is not a nuclear interaction. it is an interaction with the electron "cloud", the orbitals. I have edited Apr 20, 2017 at 5:49
• Sorry I meant to ask can one correlate the location of hits in the coordinate plane with the region around the nucleus in which the interaction took place? Apr 20, 2017 at 7:13
• if you look at the experimental set up, possibly it could be unfolded in a projection of the whole thing, using the mathematics of the hydrogen atom solutions. In the paper they dispaly the radial excitations which give rise to the orbitals and hensce to the interference image in the detector Apr 20, 2017 at 8:13