What is the significance of commuting and non-commuting operators?
If two operators <math>A</math> and <math>B</math> commute, they can be measured at the same time. They form a common basis set of the state space with eigenvectors common to both.For example, the energy and momentum eigenfunctions for the free particle both have the form of plane waves; therefore, energy and momentum operators commute and energy states are labelled with both <math>\epsilon</math> and k to uniquely define a solution.
Note: this k is different from the k used to label eingenfunctions for a perdioic potential-- for a periodic potential, states are labelled by energy and by Bloch wavenumber, implying that the Hamiltonian commutes with the Discrete Translational Operator (Lecture 7, page 3).
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Fundamental theorem from algebra
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If two operators <math>A</math> and <math>B</math> commute one can construct a basis of the state space with eigenvectors common to <math>A</math> and <math>B</math>. Conversely, if a basis set is found of eigenvectors common to both <math>A</math> and <math>B</math> then <math>A</math> and <math>B</math> commute.
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How are constants of motion used to label states?
States are labeled by specific values of their properties which do not change with time. These properties are called constants of motion. In quantum mechanics, constants of motion are physical quantities that obey the following relations.
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<math> \frac
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<math>[\hat A, \hat H] = 0</math>
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In quantum mechanics, physical properties are represented by operators. The values of properties obtained by measurements are eigenvalues of corresponding operators. The eigenvalues are used to label states.
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The fact that a property commutes with the Hamiltonian implies that a set of common eigenfunctions can be found. This means that both eigenvalues can be used to "label" the state, <math>u_
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<math>\hat H u = E u</math>
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<math>\hat A u = a u</math>
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What is the connection between symmetries and a constant of motion?
Every symmetry in a physical system implies an associated conserved quantity. Constants of motion are those physical quantities that do not vary with time. The expectation value does not change with time. Constants of motion commute with the Hamiltonian.
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<math>\frac{\partial
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<math>[\hat a , \hat H] = 0</math>
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What are the properties of "conservative systems"?
If there is no explicit time dependence of the energy function in the classical system, the total energy of the system is conserved and the system is called conservative. Practical examples can be a crystal at thermal equilibrium in the absence of time varying external fields. If there is no time dependence of the classical energy function, there will not be an explicit time dependence of the corresponding Hamiltonian operator. This leads to a very simple time dependence of the wavefunctions. The solution can be split into time-independent and time-dependent parts. See how we Schr��dinger's equation was used here
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What is the dispersion relation for a free particle, and why are E and k used as labels?
The dispersion relation forms relationships between the energy <math>E</math> and wave vector <math>k</math>. We need both to label a state, because multiple bands exist. For example, in the free particle case, the energy eigenvalues are degenerate, thus we need more than just the energy to label a state.