What the wave function is and what it is for

• The wave function is a mathematical function ${\displaystyle \psi }$ used to describe a wave. It contains all the information about the quantity that oscillates.
• The physical meaning and the units of ${\displaystyle \psi }$ vary depending on the nature of the wave itself.
• Some examples:

Note: the wave function contains direct information on the quantity that oscillates, not on the medium that may carry the oscillation. For example, for sea waves, ${\displaystyle \psi }$ describes the profile of the surface, but contains no direct information about the water. However, studying the propagation of waves in a medium often provides information about the nature of the medium. For example:

• measuring the change of direction of a light wave when it passes from air to glass tells us the value of the refractive index of glass relative to that of air (see the section refraction).
• observing how a wave is reflected or transmitted at the interface between two different substances shows directly how much the substances oppose the transmission of energy by the wave itself (see the section refraction in one dimension, in Italian)
• measuring the properties of seismic waves, such as amplitude, direction of oscillation and arrival times at the seismograph, not only shows where the epicentre of an earthquake is, but can give valuable information about the internal structure of the Earth (see the section speed of mechanical waves).
• most properties of matter at the molecular or atomic level are measured by irradiating a sample with waves (usually electromagnetic, such as visible light, ultraviolet light, X-rays, ${\displaystyle \gamma }$ rays, but also beams of electrons or neutrons) and observing the response of the sample

In the following sections we examine some properties of the wave function that do not depend on the type of wave it refers to.

The general case: travelling waves

We have said that ${\displaystyle \psi }$ is called the wave "function". A function is a correspondence between variables. What are these variables? We know that the nature of ${\displaystyle \psi }$ depends on the type of wave it describes, but the independent variables are, in most cases, always the same: the coordinates in space and time.

Let us first examine the case of waves propagating in a single dimension x.

• In general, the quantity described by the wave function varies in space and time, so ${\displaystyle \psi }$ must be a function of the position x and of the time t.

${\displaystyle \psi (x,t)}$.

• In three dimensions, instead of the coordinate x we must use the position vector r

${\displaystyle \psi (\mathbf{𝐫},t)}$.

• If the medium in which the wave propagates is itself moving, the position vector may in turn depend on time

${\displaystyle \psi (\mathbf{𝐫}(t),t)}$.

The most common case: waves that propagate without changing shape

Now assume that the wave has the property that its profile does not change during propagation. This means that, even though the disturbance of the wave travels through space, its shape does not change. Incidentally, this property, which is by no means common to all modes and media of propagation, is very important for the use of waves in telecommunications. Indeed, we can encode a signal in the waveform we transmit, but what becomes of our signal if the waveform has changed by the time the wave reaches its destination?

To find the mathematical expression of a wave whose shape does not change during propagation, we reason as follows:

At time t=0 we can take a photograph of the wave: we get the function ψ(x,0).	We wait for a time Δt and take another photograph. We get the function ψ(x, Δt).

Now, if the second photograph is nothing but a translation of the first, that is, a shift along the x axis without any change in the shape of the function, we must have

${\displaystyle \psi (x,\mathrm{\Delta }t)=\psi (x-\mathrm{\Delta }x,0)}$.

The function ${\displaystyle \psi }$ has stayed the same, but its profile has shifted to the right along the x axis by an amount ${\displaystyle \mathrm{\Delta }x}$

• If we know that the speed of translation is c, then we can relate the shift of the profile in space (${\displaystyle \mathrm{\Delta }x}$) to the time elapsed between the two photographs (${\displaystyle \mathrm{\Delta }t}$)

${\displaystyle \mathrm{\Delta }x=c\mathrm{\Delta }t}$,

and, consequently, we can obtain the expression of the wave function for any time we like:

${\displaystyle \psi (x,t)=\psi (x-ct)}$.

In the same way, a wave that propagates without changing shape towards negative x is written

${\displaystyle \psi (x,t)=\psi (x+ct)}$.

The second most common case: oscillations without propagation

The form of the wave function derived above describes a disturbance that moves through space with speed c without changing shape. What is the mathematical expression of the wave function ${\displaystyle \psi }$ if it describes an oscillation, but not a propagation from one point to another? (the section normal modes illustrates the importance of this case and contains several examples of this type of oscillation)

In this case we can reason as follows:

We take a photograph at time ${\displaystyle t=0}$ and obtain a certain profile ${\displaystyle \psi (x,0)=\varphi (x)}$	After a time ${\displaystyle t}$ the profile ${\displaystyle \varphi }$ of the wave has not moved, as it does for travelling waves, but has stayed where it was. However, it is modified by a multiplicative factor ${\displaystyle \alpha }$ that depends only on the time t. ${\displaystyle \psi (x,t)=\varphi (x)\alpha (t)}$

In this form, which describes waves without propagation, or standing waves, the space and time variables are each separated into a different function: ${\displaystyle \varphi }$ for space, and ${\displaystyle \alpha }$ for time. This is a considerable simplification for solving the wave equation.

Note that the term standing (or stationary) does not at all mean that the wave function does not depend on time; it indicates that the position of the points where the wave function is zero (called nodes) does not change over time. Indeed, if, for a certain point ${\displaystyle x_0}$, at the initial instant ${\displaystyle t=0}$ we have

${\displaystyle \varphi (x_0)=0}$,

then, at any later instant t, we also have

${\displaystyle \psi (x_0,t)=\varphi (x_0)\alpha (t)=0}$,

that is, the position of the node does not change over time.

A standing wave can also be obtained by adding two equal travelling waves propagating in opposite directions. For example

${\displaystyle \sin (kx-\omega t)+\sin (kx+\omega t)=2\sin (kx)\cos (\omega t)}$,

which has exactly the form ${\displaystyle \varphi (x)\alpha (t)}$ (see standing waves).

Spherical waves

Whenever a wave has a particular symmetry, this symmetry is reflected in its mathematical expression.

For example, we know that a wave has spherical symmetry when its profile ${\displaystyle \psi }$ does not depend on the angle from which it is viewed. We could also say that if we rotate the coordinate system, the profile does not change.

Clearly the expression

${\displaystyle \psi (\mathbf{𝐫},t)}$

still describes the spherical wave, but it is also clear that it could be greatly simplified. Indeed, we see that if we rotate the coordinates ${\displaystyle \mathbf{𝐫}\to {\mathbf{𝐫}}^{\prime }}$, the previous expression changes, and so does not make the symmetry evident.

On which variable, then, can ${\displaystyle \psi }$ depend, besides time?

It must be a variable that does not change when we rotate the coordinates. This is the case, for example, of the distance from a certain point (which may simply be the origin of the coordinates). Calling r this distance

${\displaystyle r=|\mathbf{𝐫}|}$

the spherical wave is conveniently expressed in the form

${\displaystyle \psi (r,t)}$.

If we now rotate the coordinates ${\displaystyle \mathbf{𝐫}\to {\mathbf{𝐫}}^{\prime }}$, the expression stays the same, because the distance r stays the same, ${\displaystyle r^{\prime }=|{\mathbf{𝐫}}^{\prime }|=|\mathbf{𝐫}|=r}$.

We can go further if we consider a particular type of propagation: one that takes place without any change in energy. In this case the energy that crosses, every second, a sphere of radius r centred on the source is the same for all spheres. Since the surface of the sphere, ${\displaystyle 4\pi r^2}$, grows with the square of the radius, the intensity of the wave must decrease as ${\displaystyle 1/r^2}$; and since the intensity is proportional to the square of the amplitude, the amplitude decreases as ${\displaystyle 1/r}$ (see Power of a sound source). A spherical wave that moves away from the source with speed c without changing shape, apart from this decrease of the amplitude, is therefore written

${\displaystyle \psi (r,t)=\frac{f(r-ct)}{r}}$,

where the function f describes the profile of the wave, as in the case of travelling waves in one dimension.

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