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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!
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!