Quantum Mechanics for Mathematicians
Chapter 3 Two-state Systems and SU(2) The simplest truly non-trivial quantum systems have state spaces that are in- herently two ...
and have dimensionN+1. The corresponding quantum systems are said to have “spinN/2”. The caseN= 0 is the trivial representation ...
axis, and the eigenvalues−^12 ,+^12 of this operator will be used to label the two eigenstates, so |+ 1 2 〉= ( 1 0 ) and |− 1 2 ...
and (σ 1 −iσ 2 ) ( 1 0 ) = 2 ( 0 1 ) (σ 1 −iσ 2 ) ( 0 1 ) = ( 0 0 ) (σ 1 +iσ 2 ) is called a “raising operator”: on eigenvectors ...
subgroups inside the unitary 2 by 2 matrices, but only one of them (the case j= 3) will act diagonally onH, with theU(1) represe ...
so the self-adjointness of theσjimplies unitarity ofeiθv·σsince (eiθv·σ)†= ((cosθ) 1 +i(sinθ)v·σ)† = ((cosθ) 1 −i(sinθ)v·σ†) = ( ...
The condition that the first row has length one gives αα+ββ=|α|^2 +|β|^2 = 1 Using these two relations and computing the determi ...
can instead be used. These satisfy the slightly simpler commutation relations [Xj,Xk] = ∑^3 l=1 jklXl or more explicitly [X 1 , ...
3.3 Dynamics of a two-state system Recall that the time dependence of states in quantum mechanics is given by the Schr ̈odinger ...
In this special case, the eigenvalues of the Hamiltonian areh 0 ±h 3. In the physical realization of this system by a spin^12 pa ...
Chapter 4 Linear Algebra Review, Unitary and Orthogonal Groups A significant background in linear algebra will be assumed in lat ...
operatorLonV L:v∈V→Lv∈V as multiplication by annbynmatrix: v 1 v 2 vn → L 11 L 12 ... L 1 n L 21 L ...
Definition(Dual vector space). ForVa vector space over a fieldk, the dual vector spaceV∗is the vector space of all linear mapsV→ ...
Definition(Dual or contragredient representation).The dual or contragredient representation onV∗is given by taking as linear ope ...
In the second step we are using the fact that elements of the dual basis transform as the dual representation. This is what is n ...
An inner product gives a notion of length-squared||·||^2 for vectors, with ||v||^2 =〈v,v〉 Note that whether to specify antilinea ...
For a choice of orthonormal basis{ej}, i.e., satisfying 〈ej,ek〉=δjk a useful choice of label is the indexj, so |j〉=ej Because of ...
and this kind of expression is referred to by physicists as a “completeness rela- tion”, since it requires that the set of|j〉be ...
for allv,w∈V. Such transformations take orthonormal bases to orthonormal bases, so one role in which they appear is as a change ...
so det(L) =± 1 O(n) is a continuous Lie group, with two components distinguished by the sign of the determinant: SO(n), the subg ...
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