Proof Of [Lz, Sin(2φ)] = 2iħ(sin²φ - Cos²φ) In Quantum Mechanics

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In quantum mechanics, the commutation relations between operators play a crucial role in understanding the fundamental properties of physical systems. This article delves into the proof of a specific commutation relation involving the z-component of the angular momentum operator, denoted as Lz, and the sine of twice the azimuthal angle, sin(2φ). This relation is expressed as [Lz, sin(2φ)] = 2iħ(sin²φ - cos²φ), where ħ is the reduced Planck constant and φ is the azimuthal angle in spherical coordinates. Understanding this commutation relation is essential for grasping the behavior of quantum systems with angular symmetry. In this comprehensive exploration, we will break down the proof step by step, providing a clear and detailed explanation of each stage. We will also discuss the significance of this relation in the broader context of quantum mechanics, highlighting its implications for the quantization of angular momentum and the properties of quantum states.

The primary objective of this article is to provide a rigorous and accessible proof of the given commutation relation. By meticulously working through the mathematical steps, we aim to equip readers with a solid understanding of the underlying principles and techniques. Additionally, we will explore the physical implications of this relation, shedding light on its significance in various quantum mechanical scenarios. Whether you are a student learning quantum mechanics for the first time or a researcher seeking a deeper understanding of angular momentum operators, this article offers valuable insights and a comprehensive treatment of the subject matter.

To fully appreciate the nuances of this proof, it is beneficial to have a foundational understanding of quantum mechanics, including the concepts of operators, commutators, and angular momentum. Familiarity with spherical coordinates and trigonometric identities will also be advantageous. However, we will strive to present the material in a self-contained manner, ensuring that the key concepts are clearly explained and the mathematical derivations are easy to follow. By the end of this article, readers should be able to confidently reproduce the proof and apply the result to related problems in quantum mechanics.

Background

Before diving into the proof, let's establish the necessary background information. We begin by defining the z-component of the angular momentum operator, Lz, and its representation in spherical coordinates. Angular momentum is a fundamental concept in both classical and quantum mechanics, representing the rotational inertia of a system. In quantum mechanics, angular momentum is quantized, meaning it can only take on discrete values. The operators corresponding to the components of angular momentum, such as Lz, play a crucial role in describing the rotational properties of quantum systems.

In spherical coordinates (r, θ, φ), the z-component of the angular momentum operator, Lz, is given by:

Lz = -iħ(∂/∂φ)

This operator describes the infinitesimal rotation about the z-axis. It is a Hermitian operator, which means its eigenvalues are real and correspond to the measurable values of the z-component of angular momentum. The eigenvalues of Lz are integer multiples of ħ, denoted as mħ, where m is an integer known as the magnetic quantum number. These quantized values reflect the discrete nature of angular momentum in quantum mechanics.

Next, we need to understand the function sin(2φ). This trigonometric function arises naturally in many physical contexts, particularly when dealing with systems that exhibit rotational symmetry. It can be expressed in terms of sine and cosine functions using the double-angle formula:

sin(2φ) = 2sin(φ)cos(φ)

This expression will be useful in simplifying the commutator we aim to calculate. Additionally, we will use the definitions of sin(φ) and cos(φ) in Cartesian coordinates:

sin(φ) = y / √(x² + y²)

cos(φ) = x / √(x² + y²)

These relations will help us connect the azimuthal angle φ to the Cartesian coordinates (x, y, z), which are often more convenient for calculations. Furthermore, we will utilize the following trigonometric identities:

sin²(φ) = (1 - cos(2φ)) / 2

cos²(φ) = (1 + cos(2φ)) / 2

These identities will assist us in simplifying the final expression for the commutator. Finally, we will employ the commutator identity:

[Â, B^] = B[Â, Ĉ] + [Â, B]^

This identity is a crucial tool for evaluating commutators involving products of operators. It allows us to break down complex commutators into simpler terms, making the calculation more manageable. By applying this identity, we can systematically work through the proof and arrive at the desired result. With these foundational concepts and tools in place, we are now ready to embark on the proof of the commutation relation.

Proof of the Commutation Relation

To prove the relation [Lz, sin(2φ)] = 2iħ(sin²φ - cos²φ), we will follow a step-by-step approach. First, we will explicitly write out the commutator in terms of the operators involved. Then, we will apply the commutator identity mentioned earlier to simplify the expression. Finally, we will use the definition of Lz in spherical coordinates and perform the necessary derivatives to arrive at the final result.

Step 1: Define the Commutator

The commutator of two operators, Â and B^, is defined as:

[Â, B^] = ÂB^ - B^Â

In our case, the operators are Lz and sin(2φ). Therefore, the commutator we want to evaluate is:

[Lz, sin(2φ)] = Lzsin(2φ) - sin(2φ)Lz

This expression represents the difference between the actions of the operators in two different orders. The non-commutativity of these operators reflects the fundamental uncertainty relations in quantum mechanics. The goal is to simplify this expression and show that it is equal to 2iħ(sin²φ - cos²φ).

Step 2: Apply the Product Rule for Commutators

We can rewrite sin(2φ) using the double-angle formula:

sin(2φ) = 2sin(φ)cos(φ)

Now, we can apply the commutator identity [Â, B^] = B[Â, Ĉ] + [Â, B]^, where  = Lz, B^ = sin(φ), and Ĉ = cos(φ). This gives us:

[Lz, sin(2φ)] = [Lz, 2sin(φ)cos(φ)] = 2[Lz, sin(φ)cos(φ)]

Applying the commutator identity again, we get:

2[Lz, sin(φ)cos(φ)] = 2(sin(φ)[Lz, cos(φ)] + [Lz, sin(φ)]cos(φ))

This step is crucial as it breaks down the original commutator into two simpler commutators, [Lz, cos(φ)] and [Lz, sin(φ)], which are easier to evaluate. By systematically applying the commutator identity, we can reduce the complexity of the problem and focus on calculating these individual commutators.

Step 3: Evaluate the Simpler Commutators

Now, we need to evaluate [Lz, sin(φ)] and [Lz, cos(φ)]. Recall that Lz = -iħ(∂/∂φ). Let's start with [Lz, sin(φ)]:

[Lz, sin(φ)] = Lzsin(φ) - sin(φ)Lz = -iħ(∂/∂φ)sin(φ) - sin(φ)(-iħ(∂/∂φ))

Applying the derivative, we get:

-iħ(∂/∂φ)sin(φ) = -iħcos(φ)

And:

sin(φ)(-iħ(∂/∂φ)) = -iħsin(φ)(∂/∂φ)

So,

[Lz, sin(φ)] = -iħcos(φ) + iħsin(φ)(∂/∂φ) - sin(φ)(-iħ(∂/∂φ)) = -iħcos(φ)

Next, let's evaluate [Lz, cos(φ)]:

[Lz, cos(φ)] = Lzcos(φ) - cos(φ)Lz = -iħ(∂/∂φ)cos(φ) - cos(φ)(-iħ(∂/∂φ))

Applying the derivative, we get:

-iħ(∂/∂φ)cos(φ) = iħsin(φ)

And:

cos(φ)(-iħ(∂/∂φ)) = -iħcos(φ)(∂/∂φ)

So,

[Lz, cos(φ)] = iħsin(φ) + iħcos(φ)(∂/∂φ) - cos(φ)(-iħ(∂/∂φ)) = iħsin(φ)

These calculations demonstrate the fundamental role of the partial derivative with respect to the azimuthal angle in determining the commutation relations involving Lz. The results show that the commutators [Lz, sin(φ)] and [Lz, cos(φ)] are non-zero, indicating that Lz does not commute with either sin(φ) or cos(φ).

Step 4: Substitute Back into the Original Expression

Now that we have evaluated [Lz, sin(φ)] = -iħcos(φ) and [Lz, cos(φ)] = iħsin(φ), we can substitute these results back into the expression we obtained in Step 2:

2(sin(φ)[Lz, cos(φ)] + [Lz, sin(φ)]cos(φ)) = 2(sin(φ)(iħsin(φ)) + (-iħcos(φ))cos(φ))

Simplifying, we get:

2(iħsin²(φ) - iħcos²(φ)) = 2iħ(sin²(φ) - cos²(φ))

Thus, we have shown that:

[Lz, sin(2φ)] = 2iħ(sin²(φ) - cos²(φ))

This completes the proof of the commutation relation. By carefully applying the commutator identity and evaluating the simpler commutators, we have successfully demonstrated the validity of the given relation. This result highlights the intricate interplay between angular momentum and trigonometric functions in quantum mechanics.

Significance and Implications

The commutation relation [Lz, sin(2φ)] = 2iħ(sin²φ - cos²φ) has significant implications in quantum mechanics, particularly in understanding the behavior of systems with rotational symmetry. This relation reveals the inherent non-commutativity between the z-component of angular momentum and the sine of twice the azimuthal angle, which is a manifestation of the fundamental uncertainty principles in quantum mechanics.

One of the key implications of this commutation relation is that the operators Lz and sin(2φ) cannot be simultaneously diagonalized. In other words, there are no states that are simultaneously eigenstates of both operators. This means that we cannot precisely know both the z-component of angular momentum and the value of sin(2φ) for a given quantum state. This limitation is a direct consequence of the non-zero commutator between the two operators.

The non-commutativity between Lz and sin(2φ) also has implications for the measurement process in quantum mechanics. If we measure the z-component of angular momentum, the subsequent measurement of sin(2φ) will be affected, and vice versa. This is because the act of measurement collapses the quantum state into an eigenstate of the measured operator, thereby altering the probabilities of the outcomes of subsequent measurements.

Furthermore, this commutation relation is connected to the quantization of angular momentum. The eigenvalues of Lz are quantized, taking on discrete values that are integer multiples of ħ. This quantization is a direct consequence of the commutation relations between the components of the angular momentum operator. The relation we have proven here contributes to our understanding of how angular momentum is quantized in quantum systems.

In various physical systems, the azimuthal angle φ plays a crucial role in describing the spatial orientation of particles. For example, in atoms, the electronic wavefunctions are often expressed in spherical coordinates, and the azimuthal angle determines the electron's position around the nucleus. The commutation relation between Lz and sin(2φ) thus affects the probability distribution of electrons in atoms and molecules.

Moreover, this relation is relevant in the study of quantum entanglement and quantum information processing. Entangled quantum states can exhibit correlations that are sensitive to the relative phases and orientations of particles, and the azimuthal angle is a key parameter in characterizing these correlations. The commutation relation we have discussed here provides insights into the limitations and possibilities of manipulating quantum states for technological applications.

In summary, the commutation relation [Lz, sin(2φ)] = 2iħ(sin²φ - cos²φ) is not merely a mathematical curiosity but a fundamental result with far-reaching implications in quantum mechanics. It underscores the non-classical nature of quantum systems, the limitations on simultaneous measurements, and the quantization of angular momentum. Understanding this relation is essential for anyone seeking a deeper appreciation of the quantum world.

Conclusion

In this article, we have presented a detailed and step-by-step proof of the commutation relation [Lz, sin(2φ)] = 2iħ(sin²φ - cos²φ). We began by establishing the necessary background information, including the definition of the z-component of the angular momentum operator, Lz, and the trigonometric function sin(2φ). We then meticulously worked through the proof, applying the commutator identity and evaluating the simpler commutators [Lz, sin(φ)] and [Lz, cos(φ)]. Finally, we substituted these results back into the original expression to arrive at the desired relation.

Through this process, we have demonstrated the importance of commutation relations in quantum mechanics. These relations reveal the non-classical nature of quantum systems and the limitations on simultaneous measurements. The commutation relation between Lz and sin(2φ), in particular, highlights the inherent uncertainty in knowing both the z-component of angular momentum and the value of sin(2φ) for a given quantum state.

The implications of this relation extend beyond the mathematical proof. We have discussed how this commutation relation affects the behavior of systems with rotational symmetry, the measurement process in quantum mechanics, and the quantization of angular momentum. It also has relevance in the study of quantum entanglement and quantum information processing, where the azimuthal angle plays a crucial role in characterizing the correlations between particles.

By providing a comprehensive and accessible explanation of the proof and its significance, we hope to have enhanced the reader's understanding of quantum mechanics. Whether you are a student, a researcher, or simply someone interested in the foundations of quantum physics, this article offers valuable insights into the fundamental principles that govern the quantum world. The commutation relation [Lz, sin(2φ)] = 2iħ(sin²φ - cos²φ) is a testament to the rich and intricate nature of quantum mechanics, and its study contributes to our ongoing quest to unravel the mysteries of the universe.

  • Quantum Mechanics
  • Commutation Relation
  • Angular Momentum Operator
  • Azimuthal Angle
  • Proof
  • Lz Operator
  • Sin(2φ)
  • Quantum Entanglement
  • Uncertainty Principle
  • Quantization