Question

4.21 In this problem, you will go through the steps of proving the rigorous statement of the Heisenberg uncertainty principle. Denote the expectation (average) value of an operator \(\mathbf{A}\) in a state \(|\Psi\rangle\) by \(A_{\text {avg }} .\) Thus, \(A_{\text {avg }}=\langle A\rangle=\) \(\langle\Psi|\mathbf{A}| \Psi\rangle .\) The uncertainty (deviation from the mean) in the normalized state \(|\Psi\rangle\) of the operator \(\mathbf{A}\) is given by $$ \Delta A=\sqrt{\left|\left(A-A_{\text {avg }}\right)^{2}\right\rangle}=\sqrt{\left\langle\Psi\left|\left(\mathbf{A}-A_{\text {avg }} 1\right)^{2}\right| \Psi\right\rangle} $$ (a) Show that for any two hermitian operators \(\mathbf{A}\) and \(\mathbf{B}\), we have $$ |\langle\Psi|\mathbf{A B}| \Psi\rangle|^{2} \leq\left\langle\Psi\left|\mathbf{A}^{2}\right| \Psi\right\rangle\left\langle\Psi\left|\mathbf{B}^{2}\right| \Psi\right\rangle $$ (b) Using the above and the triangle inequality for complex numbers, show that $$ |\langle\Psi|[\mathbf{A}, \mathbf{B}]| \Psi\rangle|^{2} \leq 4\left\langle\Psi\left|\mathbf{A}^{2}\right| \Psi\right\rangle\left\langle\Psi\left|\mathbf{B}^{2}\right| \Psi\right\rangle $$ (c) Define the operators \(\mathbf{A}^{\prime}=\mathbf{A}-\alpha \mathbf{1}, \mathbf{B}^{\prime}=\mathbf{B}-\beta \mathbf{1}\), where \(\alpha\) and \(\beta\) are real numbers. Show that \(\mathbf{A}^{\prime}\) and \(\mathbf{B}^{\prime}\) are hermitian and \(\left[\mathbf{A}^{\prime}, \mathbf{B}^{\prime}\right]=[\mathbf{A}, \mathbf{B}]\). (d) Now use all the results above to show the celebrated uncertainty relation Heisenberg uncertainty $$ (\Delta A)(\Delta B) \geq \frac{1}{2}|\langle\Psi|[\mathbf{A}, \mathbf{B}]| \Psi\rangle| $$ principle What does this reduce to for position operator \(\mathbf{x}\) and momentum operator \(\mathbf{p}\) if \([\mathbf{x}, \mathbf{p}]=i \hbar ?\)

Short answer

The Heisenberg uncertainty principle derived from these steps is \((\Delta A)(\Delta B) \geq \frac{1}{2}|\langle\Psi|[\mathbf{A}, \mathbf{B}]| \Psi\rangle|\). For the position and momentum operators, it becomes \(\Delta x\Delta p \geq \frac{\hbar}{2}\).

Step-by-step solution

Proving inequality for hermitian operators

We start with the operator inequality \(|\langle \Psi | \mathbf{AB} | \Psi \rangle|^2 \leq \langle \Psi | \mathbf{A}^\dagger \mathbf{A} | \Psi \rangle \langle \Psi | \mathbf{B}^\dagger \mathbf{B} | \Psi \rangle\). Now since A and B are hermitian operators, we have \( \mathbf{A}^\dagger = A, \mathbf{B}^\dagger = B \), which leads to \( |\langle \Psi | \mathbf{AB} | \Psi \rangle|^2 \leq \langle \Psi | \mathbf{A}^2 | \Psi \rangle \langle \Psi | \mathbf{B}^2 | \Psi \rangle \)

Applying the triangle inequality

With the result from the first step, we proceed to show the inequality for commutators. Given \(|\langle\Psi|[\mathbf{A}, \mathbf{B}]| \Psi\rangle|^{2} \leq4\left\langle\Psi\left|\mathbf{A}^{2}\right|\Psi\right\rangle\left\langle\Psi\left|\mathbf{B}^{2}\right| \Psi\right\rangle\), we use the triangle inequality to expand the commutator \(|\langle\Psi|[\mathbf{A}, \mathbf{B}] | \Psi\rangle|^{2} = |\langle\Psi|\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A} | \Psi\rangle|^{2}\). Simplifying this gives us the desired equality.

Defining Operators A' and B'

Next, we need to show that the operators \(A' = A - \alpha\mathbf{1}\) and \(B' = B - \beta\mathbf{1}\) are still hermitian, and their commutator is the same as \(A, B\). Since \(\alpha\) and \(\beta\) are real numbers and \(\mathbf{1}\) is the identity operator with \( \mathbf{1}^\dagger = \mathbf{1}\), A' and B' are hermitian. Their commutators remains the same as \(A, B\), because the identity operator commutes with all operators.

Deriving the Heisenberg's Uncertainty Principle

We can now proceed to derive the Heisenberg Uncertainty Principle. Using the results above and the definition of standard deviation \(σ\) we get \((\Delta A)(\Delta B) = \sqrt{\langle\Psi|A'^2|\Psi\rangle}\sqrt{\langle\Psi|B'^2|\Psi\rangle} \geq \frac{1}{2}|\langle\Psi|[\mathbf{A}, \mathbf{B}]| \Psi\rangle|\). This crucial relation is the Heisenberg Uncertainty Principle.

Applying the Uncertainty Principle to position and momentum operators

Finally, the uncertainty principle for position \(\mathbf{x}\) and momentum \(\mathbf{p}\) operators can be obtained by substituting \([\mathbf{x}, \mathbf{p}] = i\hbar\). This gives us \(\Delta x\Delta p \geq \frac{1}{2}|\langle\Psi|i\hbar| \Psi\rangle| = \frac{\hbar}{2}\), where \(\Delta x\) and \(\Delta p\) represent the uncertainties in position and momentum respectively.

Key concepts

Hermitian operators

In quantum mechanics, Hermitian operators play a pivotal role. These operators are important because they correspond to observable physical quantities, like energy and momentum. The unique property of these operators is that their eigenvalues are always real numbers. This means that any measurement associated with a Hermitian operator will yield a real, and thus physically meaningful, value.

Hermitian operators are defined by the property that the operator is equal to its own conjugate transpose. Mathematically, an operator \( \mathbf{A} \) is Hermitian if \( \mathbf{A} = \mathbf{A}^\dagger \). This is crucial because it ensures that the expected values or measured quantities in quantum mechanics are real, which aligns with observable phenomena in the physical world.

Thus, understanding Hermitian operators helps us to link mathematical models with physical reality, ensuring that the calculations we perform in quantum mechanics can be directly related to the physical observations.

Commutator

A commutator in quantum mechanics provides insight into the relationship between two operators. Specifically, the commutator of two operators \( \mathbf {A} \) and \( \mathbf {B} \), denoted as \([\mathbf{A}, \mathbf{B}]\), is defined as \( \mathbf{AB} - \mathbf{BA} \).

Understanding commutators is important because they help to determine whether two physical quantities can be simultaneously measured with certainty. If the commutator is zero, the two operators commute, meaning they can be simultaneously measured or have a well-defined value at the same time. Conversely, a non-zero commutator indicates that the operators cannot be precisely measured at the same time, leading to the inherent uncertainties characterized by the Heisenberg Uncertainty Principle.

For example, in quantum mechanics, the commutator of position and momentum is non-zero and is proportional to a fundamental constant \(i\hbar\). This non-zero commutator is fundamental to the principle of quantum uncertainty, limiting the precision with which position and momentum can be known for a particle.

Standard deviation in quantum mechanics

Standard deviation in quantum mechanics refers to the degree of deviation from the average value for a particular observable in a quantum state. Just like in classical statistics, it gives a measure of how spread out the values are in a given set.

For an operator \( \mathbf{A} \), the standard deviation \( \Delta A \) is defined as \[ \Delta A = \sqrt{\langle \Psi | (\mathbf{A} - A_{\text{avg}})^2 | \Psi \rangle} \]. This equation provides the "uncertainty" or "spread" in the measurement of the observable related to operator \(\mathbf{A}\).

The concept of standard deviation is central to the Heisenberg Uncertainty Principle. In general, a smaller standard deviation means more precise measurements. However, due to the intrinsic nature of quantum mechanics, certain pairs of observables such as position and momentum will always have a minimum level of uncertainty, no matter how precisely one tries to measure them. This is fundamentally captured by the equation \((\Delta A)(\Delta B) \geq \frac{1}{2}|\langle\Psi|[\mathbf{A}, \mathbf{B}]|\Psi\rangle|\).

Position and momentum operators

Position and momentum operators are fundamental in describing quantum systems. In the realm of quantum mechanics, the position operator \( \mathbf{x} \) and the momentum operator \( \mathbf{p} \) serve as the backbone for analyzing particle dynamics.

The position operator \( \mathbf{x} \) is related to the location of a particle in space, while the momentum operator \( \mathbf{p} \) relates to the particle's motion, taking into account both its mass and velocity. These operators behave according to specific mathematical rules and are deeply interlinked through commutation relations, such as the widely recognized \( [\mathbf{x}, \mathbf{p}] = i\hbar \).

This commutation relationship is not just mathematical formality but the heart of quantum behavior, conveying the idea encapsulated by the Heisenberg Uncertainty Principle: it is impossible to exactly know both the position and momentum of a particle at the same time. This limitation implies that at the quantum level, particles behave in ways that escape precise prediction, embracing a natural "fuzziness" rather than Newtonian determinism.