Hermitian Operators and Commutation in Quantum Mechanics

In the realm of quantum mechanics, Hermitian operators play a pivotal role in representing physical observables. This article delves into the conditions under which the multiplication of two Hermitian operators results in a Hermitian operator and whether such operators must commute.

The Importance of Hermitian Operators

In quantum mechanics, Hermitian operators are fundamental because they correspond to observables such as position, momentum, and energy. These operators have real eigenvalues, which are the possible measurement outcomes, and their eigenstates form a complete basis set for the Hilbert space. The requirement of Hermiticity ensures that these operators have these desirable properties.

The Condition for Commutation

A central concept in the theory of quantum mechanics is the commutation relation between operators. The commutator of two operators A and B is defined as:

[ [A, B] AB - BA ]

If [A, B] 0, then the operators A and B commute, meaning that the order of applying the operators does not affect the result. However, the converse is not necessarily true. In other words, two operators may commute without being Hermitian, or without their product being Hermitian.

Hermitian Product and Commutation

Given this background, we now address the question: If the multiplication of two Hermitian operators is Hermitian, does this imply that they must commute?

Let's denote the two Hermitian operators as A and B. If AB is Hermitian, then it must satisfy the condition:

[ (AB)^dagger AB ]

Since A and B are Hermitian, we have ( A^dagger A ) and ( B^dagger B ). Therefore, the Hermitian condition for AB can be rewritten as:

[ (AB)^dagger B^dagger A^dagger BA ]

Combining this with the initial condition (AB)^dagger AB, we get:

[ AB BA ]

This equation indicates that A and B must commute. Therefore, if the product of two Hermitian operators is itself Hermitian, they must commute.

Implications and Applications

The above result has significant implications in quantum mechanics. For instance, it means that if two observables represented by Hermitian operators commute, then the simultaneous measurement of these observables does not affect each other. This is a crucial property for understanding the correlations between different physical quantities in quantum systems.

Examples and Further Explorations

Consider the position and momentum operators in quantum mechanics, which are:

[ hat{x} x quad text{and} quad hat{p} -ihbar frac{d}{dx} ]

These operators do not commute, as:

[ [hat{x}, hat{p}] hat{x}hat{p} - hat{p}hat{x} -ihbar ]

However, if we consider the squared position and momentum operators, which are:

[ hat{x}^2 quad text{and} quad hat{p}^2 ]

These are Hermitian and commute, ensuring that their respective eigenvalues can be measured simultaneously without any interference.

Another example is the parity operator P and the momentum operator P, which do commute. This is crucial for many symmetries in quantum systems.

Conclusion

In summary, while the multiplication of two Hermitian operators resulting in a Hermitian operator is a sufficient condition for the operators to commute, it is not a necessary condition. The key takeaway is that when the product of two Hermitian operators is Hermitian, they must commute. This relationship is fundamental in understanding the behavior of physical observables in quantum systems.

For further reading, exploring the properties of Hermitian operators and their applications in quantum mechanics will provide a deeper insight into this elegant and powerful aspect of quantum theory.