# Intuitive understanding of a wave function

Looks like wave function is an abstract mathematical object. I was trying to see if there is a simple way to visualize this. Can someone please help with that? I was thinking may be we can think that for every object ( electron etc.) we think it to be like a number or some object assigned to every point in the space. And that number or object holds the answer to whatever is possible to know about that electron?

The most common interpretation of the wavefunction, $$\psi$$, of a particle I've come across is as follows.

If for example you have $$\psi(\textbf{x})$$, which is a wavefunction as a function of position in 3D space, then the probability of finding this particle in a tiny volume element $$\text{d}V$$ is

$$P=|\psi(\textbf{x})|^2\text{d}V$$

To find the probability that the electron is in some finite volume, $$V$$, you'd just integrate the above:

$$P=\iiint_V|\psi(\textbf{x})|^2\text{d}V$$

In other words, the (modulus squared) wavefunction indicates a probability denstity for the particle it describes.

Note also that the units of $$\psi$$ in my example above must be $$\text{m}^{-3}$$ if the probability is to be unitless.

• So if I calculate this probability for the same place every 10 minute for 24 hours will the value of probability change with time.? Also are there any restrictions on how much can it change. For example I calculate for a point and probability is 10%. Can it become like 90% in the next measurement.? Nov 15 '18 at 18:42
• It very much depends on what your system is doing. Without any information, all I can answer is "maybe". Also note that due to the face that $\psi$ is often continuous, you'd need to specify a probability within a finite region, and not a point (though you can make it as small as your experiment allows)
– Garf
Nov 15 '18 at 18:53
• Indeed, perhaps some specific examples would help? Generally the probability will depend on time, but for certain states of certain systems it can be time independent. For instance, the stationary states of a system have time independent probability densities. The falstad website has lots of nice visualizations for specific examples, see the Quantum Mechanics section here falstad.com/mathphysics.html Nov 15 '18 at 19:08