Postulates of quantum mechanics

The postulates of quantum mechanics are fundamental mathematical statements that cannot be proven. Nevertheless, they are statements that everyone agrees with.

Examples of other postulates of science and mathematics are Newton’s 2^nd law, $F=ma$ , and the Euclidean statement that a line is defined by two points respectively.

Generally, the postulates of quantum mechanics are expressed as the following 6 statements:

1) The state of a physical system at a particular time $t$ is represented by a vector in a Hilbert space.

We call such a vector, a wavefunction $\psi(x,t)$ , which is a function of time and space . A wavefunction contains all assessable physical information about a system in a particular state. For example, the energy of the stationary state of a system is obtained by solving the time-independent Schrodinger equation $\hat{H}\psi=E\psi$ .

2) Every measurable physical property of a system is described by a Hermitian operator $\hat{O}$ that acts on a wavefunction representing the state of the system.

The most well-known Hermitian operator in quantum mechanics is the Hamiltonian $\hat{H}$ , which is the energy operator. The wavefunctions that $\hat{H}$ acts on are called eigenfunctions. Eigenfunctions of a Hermitian operator in quantum mechanics are further postulated to form a complete set. Other Hermitian operators frequently encountered in quantum mechanics are the momentum operator $\hat{p}$ and the position operator $\hat{x}$ .

3) The result of the measurement of a physical property of a system must be one of the eigenvalues of an operator $\hat{O}$ .

The state of a system is expressed as a wavefunction, which can be a single basis wavefunction or a linear combination of a complete set of basis wavefunctions. Since basis wavefunctions of a Hermitian operator form a complete set, all wavefunctions can be written as a linear combination of basis wavefunctions.

Question

What about a system described by a stationary state?

Answer

The wavefunction can be expressed, though trivially, as $\psi=\sum_{n=0}^{\infty}c_n\phi_n$ , where $c_i=1$ and $c_{n\not\equiv i}=0$ (we have assumed that the spectrum is discrete, i.e. the eigenvalues are separated from one another).

It is generally accepted by scientists that the values of the coefficients $c_n$ are unknown prior to a measurement. Upon measurement, the result obtained is an eigenvalue associated with one of the eigenfunctions, and hence the phrase ‘the initial wavefunction collapses to one of the eigenfunctions’. This implies that the measurement alters the initial wavefunction such that a 2^nd measurement, if made quickly, yields that same result (this obviously refers to wavefunctions describing non-stationary states, as wavefunctions of stationary states are independent of time). If we prepare a large number of identical systems and simultaneously measure them, the values of $c_n$ are found; with $\sum_{n=0}^{\infty}\vert c_n\vert^{2}=1$ , and the expectation value of the measurements being $\langle\psi\vert\hat{O}\vert\psi\rangle$ .

4) The probability of obtaining an eigenvalue $E_i$ upon measuring a system is given by the square of the inner product of the normalised $\psi$ with the orthonormal eigenfunction $\phi_i$ .

In other words,

$\vert\langle\phi_i\vert\psi\rangle \vert^{2}=\vert\langle\phi_i\vert\sum_{n=0}^{\infty}c_n\phi_n\rangle \vert^{2}=\vert\langle c_i\phi_i\vert\phi_i\rangle \vert^{2}=\vert c_i\vert^{2}$

If the spectrum is continuous, $\psi=\int_{-\infty}^{\infty}c_n\phi_n\, dn$ and the probability of obtaining an eigenvalue in the range $dn$ is $\vert\langle\phi_n\vert\psi\rangle \vert^{2}\, dn$ .

5) The state of a system immediately after a measurement yielding the eigenvalue $E_i$ is described by the normalised eigenfunction $\phi_i$ .

We have explained in the postulate 3 that this is commonly known as the collapse of the wavefunction $\psi$ to one of the eigenfunctions $\phi_i$ . It is also known as the projection of $\psi$ onto $\phi_i$ , i.e., $\hat{P}_i\vert\psi\rangle$ ; or if $\psi$ is not normalised:

$\frac{\hat{P}_i\vert\psi\rangle}{\sqrt{\langle\psi\vert\hat{P}_i\vert\psi\rangle}}$

Question

Show that $\sqrt{\langle\psi\vert\hat{P}_i\vert\psi\rangle}$ is the normalisation constant.

Answer

To normalise a wavefunction,

$NN\int c_{i}^{*}\phi_{i}^{*}c_i\phi_i\, d\tau=1\; \; \; \Rightarrow \; \; \; N^{2}=\frac{1}{\vert c_i\vert^{2}\langle\phi_i\vert\phi_i\rangle}\; \; \; \Rightarrow \; \; \; N=\frac{1}{ c_i\sqrt{\langle\phi_i\vert\phi_i\rangle}}$

So,

$\frac{1}{ \sqrt{\langle\psi\vert\hat{P}_i\vert\psi\rangle}}=\frac{1}{ \sqrt{\langle\psi\vert c_i\phi_i\rangle}}=\frac{1}{ \sqrt{\langle c_i\phi_i\vert\psi\rangle^{*}}}=\frac{1}{ \sqrt{(c_{i}^{*}c_i\langle \phi_i\vert\phi_i\rangle)^{*}}}$

$=\frac{1}{ \sqrt{c_{i}^{*}c_i\langle \phi_i\vert\phi_i\rangle}}=\frac{1}{ c_i\sqrt{\langle \phi_i\vert\phi_i\rangle}}$

6) The time evolution of the wavefunction $\psi(t)$ is governed the time-dependent Schrodinger equation $i\hbar\frac{d}{dt}\vert\psi(t)\rangle=\hat{H}(t)\vert\psi(t)\rangle$ .

Postulates of quantum mechanics

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