- Canonical Commutation Relation in Quantum Mechanics

Introduction to Canonical Commutation Relation #

The canonical commutation relation is one of the fundamental principles in quantum mechanics, establishing the relationship between position and momentum operators.

Derivation from Energy Eigenstates #

Let’s start with energy eigenstates:

$$H|i\rangle = E_i |i\rangle, \quad H|f\rangle = E_f |f\rangle$$

Where the energy difference follows the Einstein-Planck relation:

$$E_f - E_i = \hbar \omega$$

Commutator Analysis #

Examining the matrix element of the commutator $[x,H]$:

$$\langle i|[x,H]|f\rangle = (E_f - E_i) \langle i|x|f\rangle = \hbar \omega \langle i|x|f\rangle$$

We can observe that $[x,H]$ is proportional to the time derivative of $\langle i|x|f\rangle$, which implies:

$$\frac{d\langle i|x|f\rangle}{dt} \propto \hbar \omega \langle i|x|f\rangle$$

This results in:

$$\langle i|x|f\rangle = x_0 e^{c \hbar \omega t},\quad x = x_H = e^{\frac{i}{\hbar}Ht}x_S e^{-\frac{i}{\hbar}Ht},$$

$$ x_0 = \bra{i}x_S\ket{f}$$

where $c$ is a constant.

Determining the Constant #

If $c$ is real, $\langle i|x|f\rangle$ would either approach infinity or zero, breaking time symmetry. Therefore, $c$ must be a pure imaginary number. In fact, $c = -i/\hbar$, which is from Heisenberg picture,

$$\bra{i}e^{\frac{i}{\hbar}Ht}x_0 e^{-\frac{i}{\hbar}Ht}\ket{f} = x_0 e^{-i\omega t}$$

This gives us:

$$\langle i|x|f\rangle = x_0 e^{-i\omega t}$$

And consequently:

$$\langle i|\frac{dx}{dt}|f\rangle = -i\omega \langle i|x|f\rangle$$

Deriving the Commutation Relation #

Therefore:

$$\langle i|[x, H]|f\rangle = i\hbar\langle i|\frac{dx}{dt}|f\rangle$$

Which means:

$$[x,H] = i\hbar \frac{dx}{dt} = i\hbar\frac{p}{m}$$

Substituting $H = \frac{p^2}{2m} + V(x)$:

$$[x,H] = \left[x,\frac{p^2}{2m}\right] = i\hbar\frac{p}{m}$$

This implies:

$$[x,p^2] = 2i\hbar p$$

Using the product rule for commutators:

$$p[x,p]+[x,p]p = 2i\hbar p$$

Final Result #

From this, we can derive the canonical commutation relation:

$$[x,p] = i\hbar$$

This fundamental relation is at the heart of quantum mechanics and leads to the Heisenberg uncertainty principle.