Wave Function concept

Question

What do we mean when we say wave function of electron? Does it mean wave nature of electrons? I am really confused.Without clearing this confusion i cannot proceed to molecular orbital theory.I am just a beginner .I just want a theoretical concept of wave function of an electron not involving too much of modern mathematics. What is a wave function of an orbital? is it same as wave function of an electron??

Answer 1

A wavefunction of a quantum system is the system's state written in a particular form - no more nor any less than that. Here the word state has an exactly analogous meaning to the state of a classical system insofar that the state at any time uniquely defines the state at any other time and contrariwise. The state's evolution with time in either a quantum or classical system is utterly deterministic. Any linear system - classical or quantum(see footnote 1) - has a state whose time evolution is defined by a first order equation of the form $\mathrm{d}_t\,\psi = A\,\psi$ - in the case of a quantum system the operator $A$ is skew-self-adjoint - i.e. of the form $i\,H$ where $H$ is self-adjoint - so that the state vector's $L^2$ (Pythogorean) length is preserved(see footnote 2).

The difference between a classical and quantum system is that for the classical system, knowledge of the state uniquely defines the system's behavior and what measurements we shall take from the system, whereas for a quantum system the state defines probability distributions for measurements. Moreover, straight after a measurement, the state of a quantum system is uniquely determined by the actual measurement one gets.

By dint of the experimentally observed behavior that a quantum system is in an eigenstate of the measurement operator (observable) straight after the measurement, it is convenient to write the state vector with the eigenvectors of the measurement operator in question as basis - in such a basis the measurement operator is diagonal (a "multiplication operator") with the measurement values (eigenvalues) along its diagonal.

Some people reserve the word "wavefunction" strictly for a one-particle state and when the state is written with the eigenvectors of the position operator as basis. In that case the wavefunction's square modulus can be loosely interpreted as a probability to find the "particle" in question at the position named by the eigenvalue. This is what people mean, for example, when they say that the photon has no wavefunction, because it is very difficult to define a position operator for a relativistic particle, and there is no nonrelativistic description of the photon. However, many other people (myself included) simply take the wavefunction as a synonym for "quantum state". You need to be aware of both usages in reading. I am fairly old and I get the impression - from some of the younger active members on this site- that the strict reservation of the word to mean "state in position co-ordinates" is the more modern and current one - particularly in education and so I tentatively recommend you should probably adopt this one.

Footnotes:

[1]. These comments hold in the Schrödinger picture, where measurement operators are constant and states time varying. A unitary transformation shifts us into the wholly equivalent Heisenberg picture, wherein states are constant and measurement operators time varying.

[2]. Preservation of the Pythagorean length makes sense in the Born interpretation - it's simply saying that the probability for the system to end up in some state is unity!

Answer 2

In quantum mechanics we use a wave function to describe the quantum state that an electron is in. It basically tells you where you are likely to find a particle. Notice how it is typically introduced as a function of position. That is because by taking the value at a certain location and squaring it we get the probability of finding it there.

Answer 3

The "wave function of an orbital" is just referring to "the wave function for electrons the corresponding state". When speaking of electron levels in an atom, it is commonly accepted the Copenhagen interpretation of Quantum Mechanics, where a wave function of an isolated system, contains all the information relevant for its full description.

Rigorously you would need to describe the whole atom (nucleus and electrons) with a unique wave function, but the fact that their behaviors can be decoupled, based on experimental evidences, and this means we can write separate wave functions for the nucleus and for the electrons. Even more, we generally give separate wave functions for each of the electron levels, which we distinguish by specifying the distinctive properties, which are level energy $n$, total angular momentum $l$, and angular momentum projection $m$ (Orbital names)

These wave functions contain the information of the electron in an atomic level, which are given generally in terms of the spatial coordinates, and they provide information on the spatial distribution of the electrons, i.e. the squared modulus is the probability density function of finding the electron in space, according to the Copenhagen interpretation of Quantum Mechanics.