Difference between a quantum wavefunction and a wave

Question

When reading about quantum wavefunctions, I understood that different subshells have different "shapes" of orbitals, which describe the probability density of the electron. The orbital "shape" is derived using Schrodinger's wavefunctions. However, I also know that waves are supposed to be periodic disturbances/oscillations, yet there seem to be no repeating "structures" in orbitals.

I understand that wavefunctions are used in quantum mechanics and not classical mechanics, but if they are associated with the wave behaviour of the electron, then shouldn't they be like classical waves?

What misconception do I have?

Answer 1

A wave is any function of both space and time that has one or several periodicities.

The typical wavefunction of an orbital is of the form: $$\psi(\vec{r},t)=\phi(\vec{r})e^{-\frac{iEt}{\hbar}}$$ $E$ being the energy of this orbital. This expression is a wave because:

• It's a function depending on both a space variable $\vec{r}$ and a time variable $t$.
• It has a time period given by $E=\hbar\omega$ so $T=\frac{2\pi}{\omega}=\frac{h}{E}$.

When describing the shape of this orbital, you're using $\lvert\psi\rvert^2$ which no longer depends on time. It's, arguably, not a wave, but it isn't the wavefunction either.

Answer 2

Generally, a wave is physical phenomenon, but I guess you mean its mathematical description to make it comparable to a wavefunction of which I assume that it is a solution of Schroedinger's equation or a similar equation as for instance Pauli's equation for systems with spin (well, I won't consider here Dirac's equation whose solutions are no longer a good paradigma for a wavefunction due to its relativistic character).

Waves as phenomenon appear in almost each field of physics as acoustic waves, electromagnetic waves, mechanical waves, electron waves , gravitational waves etc. What makes a wave a wave is its periodical description, but it does not need to be a harmonic wave, i.e. a sin- respectively cos-wave. It appears in classical as well as in quantum mechanics.

In contrast, a wavefunction is a purely quantum mechanical concept. In case it describes an unbound motion of a quantum mechanical particle it is mostly periodical, for instance $\Psi(\mathbf{x},t) = A e^{i\omega t -\mathbf{p}\mathbf{x}}$.

However, in a bound state, for instance inside a H-atom it is not always periodical, at least if the quantum numbers which are related with rotational symmetry $l$ and $m$ are zero. In other words Schroedinger's equation allows for non-periodical solutions.

Furthermore a wavefunction of a bound state is normalized, i.e. $\int |\Psi(x,t)|^2dx =1$. For a wavefunction of an unbound state which allows a motion until infinity the normalization is usually not possible.

Actually, in classical physics we consider mostly waves whose amplitude is an observable (exceptions exist: for instance gauge potentials) therefore the description is done by real numbers -- although for mathematical and formal convenience very often complex numbers are used of which actually the real part at the end has to be taken.

A quantum mechanical wavefunction is instrinsically be complex, so it has 2 components. However, but often in simple systems a real wavefunction can be found. A complete description of a quantum mechanical system is achieved by a wavefunction which is characterized by all quantum numbers of the system. For instance spin is a typical quantum mechanical degree of freedom. Therefore a system dealing with spin should be described by a wavefunction which incoporates spin. Further quantum numbers depending on the system have to be incoporated in the wavefunction, for instance isospin strangeness etc.

Therefore a wavefunction comprises in general much more than a mathematical description for a wave.

Answer 3

What misconception do I have?

The word "wave" is used in different ways, even in the specific context of physics.

The word "wave," on it's own, often means a function of the form: $$ f(x,t) = g(kx-\omega t)\;, $$ that is, a function that depends in a very specific way on the arguments $x$ and $t$.

For example, the function $$ f(x,t) = \sin(k x - \omega t)\;, $$ where $k$ and $\omega$ are constant, would be call a "wave." And, indeed, the sine function is periodic.

These kinds of function are solutions to a "wave equation" that looks like: $$ \frac{\partial^2 f}{\partial t^2} = v^2\frac{\partial^2 f}{\partial x^2}\;, $$ where $v^2=\omega^2/k^2$.

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When used in the content of quantum mechanics, the term "wavefunction" or "wave function," does not necessarily imply the same kind of periodicity. Any function that satisfies the Schrodinger equation (also called the "Schrodinger wave equation") $$ i\frac{\partial f(x,t)}{\partial t} = \left(-\frac{\hbar^2}{2m}\frac{\partial^2 f}{\partial x^2} +V(x,t)f(x,t)\;, \right) $$ is usually called a "wave function."

The function does not have to be periodic, although sometimes it is periodic in space and/or time.

For example, the free particle Schrödinger equation ($V=0$) can be solved by a function: $$ f(x,t) = \sin({ikx - i\frac{\hbar^2 k^2}{2m}t})\;, $$ which is periodic.

But, there are also non-periodic solutions, as you have seen. For example the hydrogen atom Schrödinger equation ($V=\frac{-e^2}{4\pi \epsilon_0 r}$) has a solution that looks like: $$ f(r,\theta,\phi,t) = \left(\cos({me^4 t/(8\epsilon_0 h^2)}) + i\sin({me^4 t/(8\epsilon_0 h^2)})\right) \frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}\;. $$

In the above function there is periodic time dependence, but not periodic spatial dependence. However, we still call the function a "wave function."