Quantum Week: An Introduction To The Mathematics Of Quantum Mechanics.

Today is the 4th day of our 'Quantum Week'. On the first day, we learnt about three major events that led to the development of quantum theory. On the second day, we saw how Rutherford and Bohr developed the atomic models and yesterday we learnt the importance of the Schrodinger Wave Equation.

There are two mathematical roads to study quantum mechanics. One is the matrix mechanics and the other is the wave mechanics. It isn't possible to cover both these in a single post, so I thought of introducing the matrix mechanics through a very famous principle of the quantum theory: Heisenberg's Uncertainty Principle. Werner Heisenberg was in fact the person behind the matrix mechanics. So today's article will be a double surprise: It will introduce matrix mechanics and also the Heisenberg's Uncertainty Principle.

The mathematical equation of the principle is something like this:

The mathematical form of Uncertainty Principle

This equations means: "You cannot simultaneously measure the position and momentum of a particle. The more precisely you know its position, less precisely you will know its momentum."

Werner Heisenberg

Heisenberg’s uncertainty principle is one such principle that truly depicts the bizarre nature of the quantum world. It is inherent in all the systems exhibiting wave like nature and it arises in quantum mechanics due to matter waves associated with all the quantum objects. Let us now see the mathematics behind this principle (and of course behind quantum mechanics).

In physics we define observable as something that can be measured. For example: position and momentum are observables in quantum mechanics. Each observable is written as an operator (self adjoint). It is no rocket science. You all have been dealing with operators since childhood. Addition, subtraction, multiplication, division etc. are all operators that act on certain numbers and produce the outcome. In calculus, d/dx is an operator, known as the differential operator, that acts on a function and produces some result. You see, operators themselves do not mean anything. We need something upon which these operators can act to produce a meaningful outcome.

Similarly in quantum mechanics, every observable is associated with an operator. The position operator, the momentum operator, the Hamiltonian (which gives the total energy of the system) etc. These operators act on a particular quantum state and produce the outcome. Now since we have turned quantum mechanics into matrix mechanics, let us see one more important mathematical property of such operators.

We all know what commutation is. Take two numbers x and y. Multiply them and you can see that xy = yx. Always, no exception. But this isn’t true in the case of matrices. If A and B are two matrices then AB is not always equal to BA. Thus the commutator of two matrices, defined by [A,B] = AB – BA, isn’t always zero as it is in the case of simple numbers. [A,B] actually quantifies how well the two observables described by these operators can be measured simultaneously

That’s it. We now have all the tools to understand what the uncertainty principle really means. If x and p represent position and momentum operator, then [x,p] = ih/2π. It is not zero. Thus you cannot simultaneously determine the position and momentum of a particle. Both are incompatible.

An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. However, the particular eigenstate of the observable A need not be an eigenstate of another observable B: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.

Matrix mechanics is one of the most followed mathematical formulation of quantum mechanics. If you want to deeply study quantum mechanics, matrices are the key.

Leave a Reply