Quantum Mechanics for Mathematicians
The operatorHhas eigenvalues that are bounded below. The Hamiltonian observableHwill have a physical interpretation in terms of ...
framework (requiring for instance the solution of a Schr ̈odinger equation in 10^23 variables). Instead of trying to resolve in ...
of norm fixed to the value 1, which fixes the amplitude ofc, leaving a remaining ambiguity which is a phaseeiθ. By the above pri ...
1.3.2 Group representations Groups often occur as “transformation groups”, meaning groups of elements acting as transformations ...
The order in which elements of the group act may matter, so the inverse is needed to get the group action property 1.2, since g ...
TakeMto be a set of 3 elementsx 1 ,x 2 ,x 3. SoF(M) =C^3. Forf∈ F(M),fis a vector inC^3 , with components (f(x 1 ),f(x 2 ),f(x ...
For a unitary representation, the matricesπ(g) take values in a subgroup U(n)⊂GL(n,C). In our review of linear algebra (chapter ...
a unitary representation ofRon the space of statesH. The corresponding self-adjoint operator is the Hamiltonian operatorH (divid ...
quantum mechanical, with no classical analog. This means that the early dis- cussion found in most physics textbooks is rather d ...
Chapter 2 The Group U(1) and its Representations The simplest example of a Lie group is the group of rotations of the plane, wit ...
When we study the harmonic oscillator (chapter 22) we will find that it has aU(1) symmetry (rotations in the position-momentum ...
such that(π|W,W)is a representation. A representation that does have such a subrepresentation is called reducible. Given two rep ...
Theorem(Schur’s lemma).If a complex representation(π,V)is irreducible, then the only linear mapsM:V →V commuting with all theπ(g ...
2.2 The groupU(1) and its representations One might think that the simplest Lie group is the one dimensional additive groupR, a ...
By theorem 2.2, sinceU(1) is a commutative group, all irreducible repre- sentations will be one dimensional. Such an irreducible ...
The representations we have found are all unitary, withπktaking values in U(1)⊂C∗. The complex numberseikθsatisfy the condition ...
value for this observable, the integerqj. A general state will be a linear super- position of state vectors from differentHqjand ...
eiθ C Cn 2 iR 1 π′(iR) U(1) U(n) π π(1) π(eiθ) π(eiθ) = eiq^1 θ 0 .. . 0 eiqnθ π′(iθ) = iq 1 θ 0 .. . 0 iqnθ ...
Recall from one of our basic axioms that time evolution of states is given by the Schr ̈odinger equation d dt |ψ(t)〉=−iH|ψ(t)〉 ( ...
of the representation is properly called an action by symmetry transformations, and that one gets conservation laws. In general ...
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