Ch 10: What's the commutator and the uncertainty principle? | Maths of Quantum Mechanics

Quantum Sense
24 Jan 202317:25
EducationalLearning
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TLDRThis video delves into the concept of the commutator, a pivotal mathematical tool for understanding observables in quantum mechanics. It explains how the commutator's non-zero value indicates that certain observables, like position and momentum, cannot have simultaneous eigenbases, leading to the Heisenberg uncertainty principle. The video also touches on the implications of commuting and non-commuting operators on the measurability of physical properties and introduces the generalized uncertainty principle, highlighting the profound impact of linear algebra on our comprehension of quantum physics.

Takeaways
  • 📌 The commutator is a crucial mathematical tool for studying observables in quantum mechanics.
  • 🔄 Two operators A and B are said to commute if AB = BA, meaning the order of the operators can be switched without affecting the result.
  • 🎯 The definition of the commutator involves a bracket and is given by the difference [AB] = AB - BA.
  • ✅ If two operators commute, their commutator is zero, and vice versa.
  • 🤹 Commuting observables allow for the simplification of expressions and the use of eigenvector properties.
  • 🌐 In quantum physics, commuting observables have orthonormal eigenbases and share a simultaneous eigenbasis.
  • 🔑 Nondegenerate eigenvectors of one commuting observable must also be eigenvectors of the other.
  • 🔍 For degenerate eigenvalues, the eigenvectors of one observable either lie within the eigenspace of the other or are orthogonal to it.
  • 🌟 The inability to have a simultaneous eigenbasis for non-commuting observables leads to the uncertainty principle.
  • 📈 The position and momentum operators have a non-zero commutator, which means they do not share an eigenbasis and cannot be simultaneously measured.
  • 🚫 The Heisenberg uncertainty principle is a direct consequence of the commutator, stating that certain pairs of physical properties, like position and momentum, cannot be precisely known at the same time.
Q & A
  • What is the definition of two operators A and B commuting?

    -Two operators A and B are said to commute if the action of operator AB is equal to the action of operator BA. In other words, you can flip the order of the two operators without any issues.

  • What is the significance of the commutator in quantum mechanics?

    -The commutator is a crucial mathematical tool in quantum mechanics because it tells us whether observables have a simultaneous eigenbasis, and therefore, whether they can be simultaneously measured. It is directly related to the uncertainty principle, as non-commuting operators cannot have a simultaneous eigenbasis, leading to inherent uncertainties in their measurements.

  • How does the commutator of two operators A and B help in simplifying expressions?

    -If two operators A and B commute, their commutator is zero. This property allows us to rearrange the order of operators in expressions without changing the result. If the commutator is non-zero, it represents a third operator C, and we can still rearrange the product of A and B by adding or subtracting this commutator C.

  • What happens when two physical observables A and B commute and have orthonormal eigenbases?

    -If two observables A and B commute and have orthonormal eigenbases, they share a simultaneous eigenbasis. This means that any eigenvector of one observable is also an eigenvector of the other, allowing for the simultaneous measurement of both observables with certainty.

  • What are the implications of non-commuting observables in terms of measurement?

    -Non-commuting observables cannot have a simultaneous eigenbasis, which means that there are no states that are eigenstates of both observables at the same time. This leads to the uncertainty principle, where the more precisely one observable is measured, the less precisely the other can be known.

  • What is the Heisenberg uncertainty principle derived from?

    -The Heisenberg uncertainty principle is derived from the generalized uncertainty principle, which focuses on the commutator of operators. It states that the product of the standard deviations of position and momentum must always be greater than zero, meaning that both quantities cannot be precisely known simultaneously.

  • What is the role of the commutator in the context of the position and momentum operators?

    -The commutator of the position and momentum operators is non-zero, indicating that these operators do not commute and do not share an eigenbasis. This is the fundamental reason why one cannot simultaneously know the exact position and momentum of a particle in quantum mechanics.

  • How does the concept of eigenspaces and eigenvectors relate to commuting observables?

    -For commuting observables, the eigenvectors of one observable must lie either inside the eigenspace of the other or be orthogonal to it. This means that the eigenvectors of one observable can be used as a basis for the other observable, leading to a shared eigenbasis and the ability to make simultaneous measurements.

  • What is a stronger form of the uncertainty principle mentioned in the script?

    -The Maccone–Pati uncertainty relation is mentioned as a stronger form of the uncertainty principle. It involves the sum of variances and includes more complex expressions involving the commutator and other operators.

  • How does the commutator relate to the eigenvalues and eigenvectors in the case of degenerate eigenvalues?

    -In the case of degenerate eigenvalues, the commutator helps to show that the eigenvectors of one observable must either lie within the eigenspace of the other or be orthogonal to it. This leads to the conclusion that eigenvectors can be found that are eigenvectors of both observables, even in the presence of degeneracy.

  • What is the significance of the commutator in the overall framework of quantum mechanics?

    -The commutator is significant in quantum mechanics because it underpins the fundamental principles of the theory, such as the uncertainty principle. It dictates the possibility of simultaneous measurements of observables and is central to understanding the behavior of quantum states under measurement and observation.

Outlines
00:00
🔍 Introduction to Commutators and Quantum Uncertainty

This paragraph introduces the concept of commutators, a fundamental mathematical tool in quantum mechanics. It explains that if two operators A and B commute, their action can be interchanged without affecting the result. The importance of commutators is highlighted by their connection to uncertainty relations in quantum mechanics. The video aims to build intuition around why uncertainty relations, such as the Heisenberg uncertainty principle, exist due to the non-commutative nature of certain observables.

05:03
📐 Exploring Commuting Observables and Eigenbases

The second paragraph delves into the properties of commuting observables and their shared eigenbases. It discusses the implications of nondegenerate and degenerate eigenvalues, explaining how eigenvectors of one observable can also be eigenvectors of another if they commute. The paragraph uses the example of a twofold degeneracy in R3 to illustrate how eigenvectors within an eigenspace remain within it when acted upon by a commuting observable. It also touches on the concept of eigenvectors being orthogonal to the eigenspace when considered in the context of degenerate eigenvalues.

10:08
🌀 The Connection Between Commutators and Eigenvectors

This paragraph establishes the connection between the commutator of two operators and their eigenvectors. It explains that if two operators A and B commute, they share a simultaneous eigenbasis, meaning there exists a set of vectors that are eigenvectors for both operators. This has significant implications in physics, as it allows for the simultaneous measurement of observables like momentum and energy. The paragraph also discusses the implications of non-commuting operators, which cannot have a simultaneous eigenbasis, leading to the inherent uncertainty in measuring certain pairs of observables, such as position and momentum.

15:11
🌟 The Significance of the Commutator in Quantum Mechanics

The final paragraph emphasizes the profound impact of the commutator on our understanding of quantum mechanics. It connects the mathematical concept of non-commutativity to the physical reality of uncertainty principles, such as the Heisenberg uncertainty principle. The paragraph also mentions the generalized uncertainty principle and the Maccone–Pati uncertainty relation, both of which involve the commutator and further illustrate its importance in quantum theory. The video concludes by highlighting the beauty of how linear algebra principles underpin the fundamental aspects of quantum mechanics, including the limits of our knowledge about the universe.

Mindmap
Keywords
💡Commutator
The commutator is a mathematical tool used to determine whether two operators A and B can be interchanged without changing the result of their combined action. In the context of quantum mechanics, it is crucial for understanding the uncertainty principle. If the commutator of two observables is nonzero, they do not share a simultaneous eigenbasis, meaning certain pairs of physical properties, like position and momentum, cannot be precisely known at the same time. The video script emphasizes the importance of the commutator in quantum physics and its role in the Heisenberg uncertainty principle.
💡Observables
Observables in quantum mechanics refer to physical quantities that can be measured. They are represented by operators, and the possible outcomes of measurements are the eigenvalues of these operators. The video script discusses how the commutativity of observables determines whether they can be measured simultaneously. If two observables commute, they share a simultaneous eigenbasis, allowing for simultaneous measurement; if they do not commute, they cannot be precisely measured at the same time.
💡Eigenvectors
Eigenvectors are vectors that, when acted upon by a linear operator, only change by a scalar factor (the eigenvalue). In quantum mechanics, eigenvectors correspond to states that have a definite value for the observable they are associated with. The video script explains that if two observables commute, their eigenvectors can form a simultaneous eigenbasis, meaning a single set of vectors that can represent the states of both observables simultaneously.
💡Eigenvalues
Eigenvalues are the scalar factors by which an eigenvector is scaled when a linear operator is applied to it. In the context of quantum mechanics, eigenvalues represent the possible outcomes of measurements for a given observable. The video script relates eigenvalues to the concept of observables and their measurement, emphasizing that eigenvectors with the same eigenvalue form a degenerate eigenspace.
💡Uncertainty Relations
Uncertainty relations, such as the Heisenberg uncertainty principle, are fundamental concepts in quantum mechanics that state there are limits to the precision with which certain pairs of physical properties can be known simultaneously. The video script connects the concept of uncertainty relations to the commutator of operators, explaining that non-commuting observables lead to uncertainty in their simultaneous measurement.
💡Eigenbasis
An eigenbasis is a set of eigenvectors that spans the entire vector space of a linear operator. In quantum mechanics, an eigenbasis for an observable represents all possible states that can be associated with that observable. The video script discusses the importance of a simultaneous eigenbasis for commuting observables, which allows for simultaneous measurement of the observables.
💡Noncommuting Operators
Noncommuting operators are operators that do not satisfy the commutation relation, meaning the order in which they are applied affects the result. In quantum mechanics, this property is significant because it leads to the impossibility of simultaneously measuring certain pairs of observables with arbitrary precision. The video script highlights that noncommuting operators are the reason why the Heisenberg uncertainty principle exists.
💡Linear Operators
Linear operators are mathematical functions that act on vectors in a vector space, transforming them in a way that is consistent with the operations of vector addition and scalar multiplication. In quantum mechanics, observables are represented by linear operators, and their eigenvectors and eigenvalues provide information about the possible outcomes of measurements. The video script uses the properties of linear operators to discuss the behavior of eigenvectors under the action of commuting and noncommuting operators.
💡Hermitian Operators
Hermitian operators, also known as self-adjoint operators, are linear operators that have the property of being equal to their own conjugate transpose. In quantum mechanics, observables are represented by Hermitian operators, which ensures that the eigenvalues are real numbers, corresponding to the possible measurements. The video script mentions Hermitian operators in the context of taking the inner product with an eigenvector, which is crucial for understanding the behavior of operators acting on vectors orthogonal to an eigenspace.
💡Degenerate Eigenvalues
Degenerate eigenvalues occur when two or more eigenvectors of an operator correspond to the same eigenvalue. This situation leads to the formation of an eigenspace, which is a subspace spanned by the eigenvectors associated with that eigenvalue. The video script discusses the concept of degenerate eigenvalues and eigenspaces in the context of quantum mechanics, explaining how they relate to the simultaneous measurement of commuting observables.
💡Superposition
In quantum mechanics, a superposition refers to a state that is a linear combination of multiple eigenstates, each corresponding to a different eigenvalue of an observable. The principle of superposition is fundamental to quantum mechanics, allowing for the existence of states that are not eigenstates of a given observable. The video script explains that when measuring non-commuting observables, a quantum state can be in a superposition of different eigenstates of one observable after measuring the other.
Highlights

The introduction of the commutator as a crucial mathematical tool in studying observables in quantum mechanics.

Defining operators A and B as commuting if AB equals BA, allowing the order of the operators to be switched without issue.

The motivation for the commutator stems from the desire to switch the order of operators like energy and momentum in quantum calculations.

The commutator of two operators A and B is defined as a bracketed difference, [A, B] = AB - BA.

If two operators commute, their commutator is zero, and if the commutator is zero, the operators imply that they commute.

The general non-commutativity of operators leads to different transformations when the order of the operators is switched.

The derivation of a property of commuting observables where nondegenerate eigenvectors of one must also be eigenvectors of the other.

The exploration of degenerate eigenvalues and eigenspaces, showing that eigenvectors of B must lie either inside or orthogonal to the eigenspace of A.

The conclusion that commuting observables A and B share a simultaneous eigenbasis, which is a significant result in quantum physics.

The practical implication in physics where commuting observables like momentum and energy allow for simultaneous measurement of their eigenvalues.

The contrast between commuting and non-commuting operators, where non-commuting operators cannot have a simultaneous eigenbasis.

The fundamental connection between the commutator and the Heisenberg uncertainty principle, which limits simultaneous knowledge of certain pairs of observables.

The position and momentum operators are an example of non-commuting operators, leading to the impossibility of simultaneously determining their eigenvalues with certainty.

The significance of the commutator in quantum mechanics, being responsible for the 'weirdness' and foundational principles of the theory.

The mention of the generalized uncertainty principle, which includes more than one uncertainty relation, with the commutator as a central component.

The Maccone–Pati uncertainty relation as an example of a stronger form that includes the sum of variances and highlights the importance of the commutator.

The final takeaway that the commutator's non-equality prevents the simultaneous measurement of certain observables, such as position and momentum.

A preview of the upcoming discussion on unitary operators in quantum mechanics and their role in the derivation of the Schrödinger equation.

Transcripts
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