A layman's introduction to Quantum Mechanics

Physics, like any other branch of science, is an aspiration of many to understand the world around them. When you see a car coming towards you, you try your best to dodge the incoming car so as to avoid a collision as any sane minded individual would. You act in that manner because you know that if you don't, then that car is going to hit you and hurt you, a collision is inevitable. You are certain about that. Things are a bit different in the atomic level, or atleast that's what the copenhagon interpretation of quantum mechanics says.

As a child riddled with curiosity, I too have had the misfortune of dwelling a bit too much in the realm of "pop-culture science", unfortunately, while that may provide you with the feeling of "knowing" something, the true satisfaction of actually understanding something and therefore by virtue of which, knowing it, can only be provided by a rigorous development of physics using mathematics.

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. - Richard Feynmann, Nobel Laureate in Physics

Quantum Mechanics is a theory that aims to describe the behaviour of particles typically at the atomic level. Like any other theory, it has some postulates of its own that cannot be questioned for they are assumptions not claims.

Postulate 1

Physical States are represented by unit vectors in the Hilbert Space, \(\mathcal{H}\).

A typical representation of a vector in a \(\mathcal{H}\) is given by \[ \ket {\psi} \] (also known as 'ket')

The corresponding vector in the dual space is represented as \[ \bra {\psi} \] (also known as 'bra')

note - The above notation (bra-ket notation) is also referred to as the dirac-notation, popularized by Paul Dirac.

A Hilbert Space, \(\mathcal{H}\) is a vector space equipped with an inner product.

An inner product between 2 vectors say \(\ket {\psi}\) and \(\ket {\phi}\) is given as \[ \braket{\psi|\phi} \]

Some trivial properties and facts:
- \(|\braket{\psi|\phi}| \geq 0\)
- \(\mathcal{H}\) is a linear vector space
- \(\braket{\psi|\phi}^* = \braket{\phi|\psi}\)
- \(|\braket{\psi|\phi}| = 1\) (Since the states are unit vectors)

Postulate 2

A physical quantity or a physical property that can be measured is called as an "Observable". Examples include position, momentum, energy, etc. In quantum mechanics, observables are are linear , self-adjoined operators that are applied to the state of the system i.e. in a d-dimensional \(\mathcal{H}\), an observable is a dxd matrix that transforms a vector to another vector in the same space.

An observable acts on a ket (\(\ket {\psi}\)), i.e. \[ \hat{A} \ket {\psi} = \ket {\phi} \] where \(\ket {\phi}\) is some other vector in \(\mathcal{H}\).

An observable also acts on a bra (\(\bra {\psi}\)) in much the same way, i.e. \[ \bra {\psi} \hat{A'} = \bra {\phi} \] where \(\bra {\psi}\) is a vector in the dual space and \(\bra {\phi}\) is some other vector in the same dual space.

We know that if an operator, \(\hat{A}\) satisfies the following equation: \[ \hat{A} \ket {\psi} = \lambda \ket {\psi} \] then, \(\lambda\) is an eigenvalue of \(\hat{A}\) and \(\ket {\psi}\) is the eigen-vector corresponding to the eigen-value \(\lambda\) of \(\hat{A}\).

There's something called as Spectral Theorem, one of the implications of which is the following:

Any normal matrix, A, can be expressed as: \[ A = \sum_{n} \lambda_n P_n \] where, \(\lambda_n\) = eigen value of A

\(P_n\) = Projection matrix onto the eigenvector \(\ket {\psi_n}\) corresponding to \(\lambda_n\)

note = A normal matrix is a matrix A such that A*A = AA* thus, it is a more general set than that of the self-adjoint (Hermition matrix) that we are concerned with.

Some trivial facts and properties -
- \(P_m P_n = \delta_{mn} P_n\), where \(\delta_{mn} = \mathbb{1}_{(m=n)}\)
- \(P_n ^*= P_n\)
- \(\sum_{n} P_n = I\)

Postulate 3

The very act of observing disturbs the system. - Werner Heisenberg, Nobel Laureate in Physics

The wave function collapse -

The copenhagon interpretation of quantum mechanics says that when you make a measurement of the observable in question, the result that you will get will be one of the possible eigen values associated with the observable.

The result i.e. \(\lambda_i\) that would occur in a measurement will be probabilistic in nature and upon measurement, the wave function of the system will collapse to the eigenvalue, \(\ket {\psi_i}\) associated with the result of the measurement, i.e. \(\lambda_i\).

Unfortunately, no one can predict with absolute certainty the eigenstate to which the wave function will collapse…well unless the initial state is already an eigenstate. What one can do, however, is provide the probabilities with which the those eigenvalues, \(\lambda_i\) occur in the measurement thus giving a probability of the wave function collapsing to a particular eigenstate.

Say the state, \(\ket {\psi}\) has gone to one of the eigenvectors, say \(\ket {\psi_n}\), then \[ \ket {\psi} \rightarrow P_n \ket {\psi} \]
Staying true to the postulate, the post-measurement state would be nothing but normalized \(\ket {\psi_n}\) i.e.

\[ \frac{P_n \ket {\psi}}{\sqrt{\braket{\psi|(P_n)^2|\psi}}} \] which is just,

\[ \frac{P_n \ket {\psi}}{\sqrt{\braket{\psi|P_n|\psi}}} \]

Postulate 4

One might wonder, what happens to the state \(\ket {\psi}\), when its not being measured? Well, it obviously doesn't collapse but evolves with time governed by an equation given by Erwin Schrodinger known as the Schrodinger's equation.

\[ i\hbar \frac{\partial{\ket {\psi}}}{\partial t} = H \ket {\psi} \]

H = Hamiltonian Operator applied to the state, \(\ket {\psi}\)

The solution to this equation is given by,

\[ \ket {\psi(t)} = \mathcal{U}(t) \ket {\psi(0)} \]

where \(\mathcal{U}(t)\) is a unitary operator, the formal solution of which is given by,

\[ \mathcal{U}(t) = exp(\frac{-iHt}{\hbar}) \]