Question

Is it possible to simultaneously know the angular orientation of a molecule rotating in a two-dimensional space and its angular momentum? Answer this question by evaluating the commutator \([\phi,-i \hbar(\partial / \partial \phi)]\)

Short answer

No, it is not possible to simultaneously know the angular orientation of a molecule rotating in a two-dimensional space and its angular momentum. The commutator \([\phi, -i\hbar(\partial / \partial \phi)]\) is calculated to be non-zero, as \(-i\hbar\phi + i\hbar(2\phi) \neq 0\). This result is in accordance with the Heisenberg Uncertainty Principle, which states that if the commutator of two observables is not zero, they cannot be known simultaneously with arbitrary precision.

Step-by-step solution

Definition of the Commutator

In general, the commutator \([A,B]\) of two operators \(A\) and \(B\) is defined as \([A,B] = AB - BA\). In this case, we have \(A=\phi\) (the angular orientation) and \(B=-i\hbar(\partial / \partial\phi)\) (the angular momentum operator).

Calculating the First Term AB

We will now calculate the first term \(AB\), which is the multiplication of \(\phi\) with \(-i\hbar(\partial / \partial\phi)\). Using the multiplication rule and the fact that \(\phi\) is a function of \(\phi\), we get: \(AB = \phi(-i\hbar(\partial / \partial\phi)) = -i\hbar\phi(\partial / \partial\phi).\)

Calculating the Second Term BA

Now we will calculate the second term \(BA\), which is the multiplication of \(-i\hbar(\partial / \partial\phi)\) with \(\phi\): \(BA = -i\hbar(\partial / \partial\phi)\phi.\)

Evaluate the Commutator

Now we will find the difference between \(AB\) and \(BA\), as per the definition of the commutator. \[ [\phi, -i\hbar(\partial / \partial \phi)] = AB - BA = -i\hbar\phi(\partial / \partial\phi) - (-i\hbar(\partial / \partial\phi)\phi). \]

Simplify the Commutator

Now we will simplify the commutator by calculating the derivatives: \( [\phi, -i\hbar(\partial / \partial \phi)] = -i\hbar\phi(\partial / \partial\phi) - (-i\hbar(\partial / \partial\phi)\phi) \) \( = -i\hbar\phi(\partial\phi / \partial\phi) + i\hbar(\partial(\phi\phi) / \partial\phi) \) \( = -i\hbar\phi + i\hbar(2\phi). \)

Conclusion

The commutator is non-zero: \( [\phi, -i\hbar(\partial / \partial \phi)] = -i\hbar\phi + i\hbar(2\phi) \neq 0. \) This implies that it is not possible to simultaneously know the angular orientation of a molecule rotating in a two-dimensional space and its angular momentum with arbitrary precision, according to the Heisenberg Uncertainty Principle.

Key concepts

Commutator in Quantum Mechanics

In quantum mechanics, a commutator is an important tool used to measure the degree to which two operators do not commute with each other. This concept is notably different from multiplication in standard arithmetic. The commutator of two operators, \([A, B]\), is defined as the difference between the application of these operators in sequence: \([A, B] = AB - BA\). This essentially indicates how much applying the operations in one order differs from doing them in the reverse order.

Commutators help us understand various quantum phenomena. For instance, they are used to determine whether certain properties of quantum systems can be simultaneously known or measured. If a commutator between two observable operators is zero, these observables can be measured exactly at the same time. If the commutator is non-zero, as in our case with angular orientation and angular momentum, it indicates a fundamental quantum restriction on precise measurement of both at the same time.

Angular Momentum in Quantum Systems

Angular momentum in quantum mechanics is a critical concept related to the rotational motion of particles. It is quantized, meaning it can take on discrete values rather than a continuous range. The angular momentum operator, often denoted as \(L\), behaves quite differently from classical momentum and is subject to the rules of quantum operators.

In our two-dimensional case, the angular momentum operator for a molecule rotating about an axis is represented by \(-i\hbar(\partial / \partial \phi)\). Here, \(\hbar\) is the reduced Planck constant, highlighting the quantum nature of this concept. This operator plays a pivotal role in determining the eigenstates and eigenvalues, which define possible observed values of angular momentum in a given quantum state.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a foundational theory in quantum mechanics, stating that there is a limit to how precisely we can know pairs of certain properties of a particle, like position and momentum, simultaneously. This principle is directly connected to the non-zero commutators.

In the context of angular momentum and angular orientation, the non-zero result of the commutator \([\phi, -i\hbar (\partial / \partial \phi)]\) implies that we cannot precisely determine both the angle and the angular momentum simultaneously for a rotating molecule. This uncertainty is not due to measurement limitations, but a fundamental aspect of nature as dictated by quantum laws.

Angular Orientation

Angular orientation, in a rotating system, refers to the specific angle at which the system axis is aligned relative to some reference position. In quantum mechanics, it is represented as a variable, often denoted by \(\phi\) in two-dimensional spaces.

Understanding angular orientation in conjunction with angular momentum in quantum systems becomes complex due to their intrinsic quantum properties. In our context, the angular orientation \(\phi\) interacts with the angular momentum operator in such a way that their commutator is non-zero. This indicates that if you were to measure the angular orientation with high precision, the angular momentum would become correspondingly uncertain and vice versa, showcasing the entwined relationship dictated by quantum mechanics.