What is the uncertainty principle

Definition/Summary

This is a derivation of the Uncertainty Principle based on the properties of non-commuting Hermitian operators.

Equations

$$\langle (\Delta A)^2 \rangle \langle (\Delta B)^2 \rangle \geq \frac{1}{4} |\langle [A,B] \rangle | ^2$$

$$\langle (\Delta x_i)^2 \rangle \langle (\Delta p_i)^2 \rangle \geq \frac{1}{4} \hbar ^2$$

Extended explanation

Let A, B be a pair of operators. We define $\Delta A \equiv A - \langle A \rangle I$, where the expectation value of A with respect to some state $|a \rangle$, is defined as $\langle A \rangle = \langle a | A | a \rangle$. This number tells you what A will be measured as, on average, over several repeated measurements performed on the system, when prepared identically.

Now, we define an important quantity: the variance or mean square deviation, which is $\langle ( \Delta A ) ^2 \rangle$. This quantity is no different from the variance in any statistical collection of data. Plugging in from above

$$\langle ( \Delta A ) ^2 \rangle = \langle (A - \langle A \rangle I)^2 \rangle = \langle A^2 \rangle - \langle A \rangle ^2 ~~~-~(1)$$

Let $|x \rangle$ be any arbitrary (but normalized) state ket. Let

$$|a \rangle = \Delta A ~ |x \rangle$$
$$|b \rangle = \Delta B ~ |x \rangle$$

First we apply the Cauchy-Schwarz inequality (which is essentially a result that is two steps removed from saying that the length of a vector is a positive, real number): $\langle a |a \rangle \langle b |b \rangle \geq | \langle a |b \rangle |^2$, to the above kets (keeping in mind that $\Delta A~, ~\Delta B$ are Hermitian), giving

$$\langle (\Delta A)^2 \rangle \langle (\Delta B)^2 \rangle \geq |\langle \Delta A \Delta B \rangle | ^2 ~~~-~(2)$$

Next we write

$$\Delta A \Delta B = \frac{1}{2}(\Delta A \Delta B - \Delta B \Delta A) + \frac{1}{2}(\Delta A \Delta B + \Delta B \Delta A) = \frac{1}{2}[\Delta A, \Delta B] + \frac{1}{2}\{ \Delta A, \Delta B \} ~~~-~(3)$$

Now, the commutator

$$[\Delta A,~ \Delta B] = [A - \langle A \rangle I,~B - \langle B \rangle I] = [A,B] ~~~-~(4)$$

And notice that $[A,B]$ is anti-Hermitian, giving it a purely imaginary expectation value. On the other hand, the anti-commutator $\{ \Delta A,~ \Delta B \}$ is clearly Hermitian, and so, has a real expectation. Thus

$$\langle \Delta A \Delta B \rangle = \frac{1}{2}\langle [A,B] \rangle + \frac{1}{2} \langle \{ \Delta A,~ \Delta B \} \rangle ~~~-~(5)$$

Since the terms on the RHS are merely the real and imaginary parts of the expectation on the LHS, we have

$$| \langle \Delta A \Delta B \rangle |^2 = \frac{1}{4}| \langle [ A,B] \rangle |^2 + \frac{1}{4} | \langle \{ \Delta A,~ \Delta B\} \rangle |^2 \geq \frac{1}{4}| \langle [A,B] \rangle |^2~~~-~(6)$$

Using the result of (6) in (2) gives :

$$\langle (\Delta A)^2 \rangle \langle (\Delta B)^2 \rangle \geq \frac{1}{4} |\langle [A,B] \rangle | ^2 ~~~-~(7)$$

The above equation (7), is the most general form of the Uncertainty Relation for a pair of hermitian operators. So far, it is nothing more than a statement of a particular property of certain specifically constructed hermitian matrices.

Notice that if the operators A, B commute (ie: [A,B] = 0), then the product of the variances vanish, and there is no uncertanity in measuring their observables simultaneously. It is only in the case of non-commuting (or incompatible) operators, that you see the more popular form of the Uncertainty Principle, where the product of the variances does not vanish.

Specifically, in the case where $A = \hat{x_i}~,~~B = \hat{p_i}$, we use the commutation relation:

$$[\hat{x_i},\hat{p_i}] = i \hbar ~~~-~(8)$$

This equation follows from the definition of the quantum mechanical momentum operator, which is constructed upon the following two observations:

(i) In classical mechanics, momentum is the generator of infintesimal translations. The infinitesimal translation operator, $\tau (d \mathbf{x})$, defined by $\tau (d \mathbf{x}) |\mathbf{x} \rangle \equiv |\mathbf{x} + d \mathbf{x} \rangle$ can be written as

$$\tau (d \mathbf{x}) = I - i\mathbf{K} \cdot d \mathbf{x}$$

(ii) K is an operator with dimension length -1, and hence, can be written as $\mathbf{K} = \mathbf{p} / [action]$. The choice of this universal constant with dimensions of action (energy*time) comes from the de broglie observation $k = p/ \hbar$. So, writing $\tau (d \mathbf{x}) = I - i\mathbf{p} \cdot d \mathbf{x} /\hbar$ leads to the expected commutation relation , $[\hat{x_i}, \hat{p_i} ] = i \hbar$.

Plugging this into (7) gives the correct expression for the HUP:

$$\langle (\Delta x_i)^2 \rangle \langle (\Delta p_i)^2 \rangle \geq \frac{1}{4} \hbar ^2$$

It is this expression that is often popularized in the (somewhat misleading) short-hand: $\Delta x \Delta p \geq \hbar/2$

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

Jared Edington