Mathematics Behind Heisenberg's Uncertainty Principle.

They say quantum mechanics is weird. What is weirder is the mathematical formulation of this subject. The mathematics of quantum mechanics can be divided into two forms: wave mechanics and matrix mechanics. Out of these two, the latter is very compact and elegant but at the same time, it is difficult to comprehend and make physical interpretation about it. In the 15th article of Month of Equations series, I will talk about the mathematics behind the Heisenberg's uncertainty principle. I want to introduce you to the importance of matrices in quantum mechanics. So let us start.

Heisenberg's uncertainty principle

Uncertainty Principle
The Heisenberg Uncertainty Principle

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.

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 understand the mathematics behind this principle using matrix mechanics.

Operators In 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.

Also watch: How was quantum mechanics born?

For example, the operator for total energy of the system is known as the Hamiltonian H. When this H acts on a wavefunction, it produces the energy of the system i.e. Hψ = Eψ where ψ is the wavefunction and E is the energy eigen value. This is in fact the famous Schrodinger wave equation of quantum mechanics. We cannot cancel out the two ψs because the one on the L.H.S. is being acted on by an operator.

Similarly there are operators for other observable quantities: position, momentum, time reversal 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.

The Commutator in quantum mechanics

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.

Uncertainty principle in Commutator form

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. [x,p] = ih/2π is another form in which Uncertainty principle can be written.

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.

Author's Message

For a physicist, quantum mechanics is a very important subject. All of nuclear and particle physics is nothing but quantum mechanics. But it is very important to study the mathematics first. For matrix mechanics, it is important to learn the concept of matrices and for wave mechanics, the topic of special functions is crucial. Don't get carried away by what pop science tells about quantum mechanics. There is a quote on quantum mechanics by Richard Feynman, "I can safely say that no one really understands quantum mechanics." People take such quotes too seriously. It is not true. Had it been true, mankind would not have been able to develop the technology that uses quantum mechanics. I often say, technology is the proof that we understand the theory well.

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