There are a few postulates (as we like to call them) in Quantum Mechanics that can be treated as the “definitions” of the basic concepts in Quantum Mechanics. Some of these are new, some of these are familiar but expressed in a strange way. Here they are:
The state of a particle is represented by a normalized vector Ψ (in Hilbert space) which contains all that can be known about that particle. We call it the wave function.
In classical mechanics, we deal with the motion of a particle. We also understand that if we know the present position, velocity and net force acting on a particle, we can predict its future position and velocity and may even calculate their value in the past. Newton’s laws are sufficient there to tell you exactly how the particle moves, has moved and will move(i.e. the trajectory in the configurational space); provided you know the forces and present position and velocity. In quantum mechanics we seldom deal with position variable and rather deal with the wave function.
Here, the wave function is a quantity that has all the information about the particle at an instant. e.g. the function Ψ1 may represent the system in state 1 while Ψ2 represent it in state The energy of the system in state 1 would then be called E1 and that in state 2 would be E2. We don’t have any classical analog of this, we can only say that it is derived from the fact that particles can behave as waves. We may call it “matter wave” but we can’t connect it with anything physical that can be measured directly. However, you can make calculations with it and derive results for energy, position, momentum etc. that can be measured.
Every observable quantity has been assigned an “operator” which works on the wave function and derives value of that observable.
An operator is something that works on the wave function to yield values. It can be thought of as a symbol or label that tells you what calculations to perform with the wave function. e.g. we denote differentiation by the operator d/dx and write diferentiation of a function f(x) as
(d/dx)(f(x)) or, df(x)/dx
Physical quantities like Energy, Momentum etc. which we have intuitive notions about, are somewhat the same in the quantum world. To calculate the value of a physical quantity q, you need the associated operator Ô which then acts on Ψ to yield value of q.
ÔΨ = qΨ
Or, if there are more than one possible values(qi) associated with each of the possible Ψi then we call them eigenvalues of the operator Ô and the wave functions associated with them (each of the possible Ψi ) to be eigenfunctions of Ô:
ÔΨi = qiΨi
The operator associated with a physically measurable quantity will be Hermitian.
By Hermitian operators, we mean that the eigenvalues are real(there are other properties of course, but this is the most relevant one at this point). This is so because if you measure something that has a physical existence, it has got to be real. On the other hand, there are quantities that are complex and thus contain imaginary parts as well (the wave function itself is an example) but the quantities we usually deal with in QM are all associated with Hermitian operators.
The set of all eigenfunctions of Ô will form a complete set of linearly independent functions.
The eigenfunctions (Ψs) associated with a Hermitian operator Ô form a set of functions. This set is complete in the sense that any other function f can be expressed as a weighted sum of these:
f = ∑cnΨn
Perhaps a little analogy can make things clearer. Remember the cartesian coordinates? The x, y and z there span the 3-dimensional space and any vector can be expressed as a set of components each associated with x, y and z direction. Here, the task of unit operators x, y and z is performed by various Ψs (can be expressed as Ψ1, Ψ2 etc.) which are all orthogonal to each other and span the whole Hilbert space (at least in the discrete/finite dimensional case).
For a system described by a given wave function, the expectation value of any property q can be found by performing the expectation value integral with respect to that wave function.
The term expectation value is just a way of denoting the “average value” of the observable. More on this will require some more understanding of how wave functions behave hence we treat it later. Mathematically, we express it as:
<Ô> = ∫ Ψ*(x) Ô Ψ(x) dx
The time evolution of the wave function is governed by the time dependent shrödinger equation.
The Schrödinger equation plays a vital role in QM, just like Newtonian Laws do in classical mechanics. It tells you how the wave function evolves. It sets the rules which give you the past and future state of the system in terms of its state function Ψ, provided you know exactly what Ψ is at the present instant.
Introduction to Quantum Mechanics- David J. Griffiths