QuantumPhysics.dvi

4.3 Expectation values and quantum fluctuations If|φ〉is an eigenstate of an observableAwith eigenvaluea, then the probability fo ...

The degree to which this is possible is expressed by the Heisenberg uncertainty relations. We begin by defining the following tw ...

4.5 Complete sets of commuting observables For many simple quantum systems, the energy of a state completelyspecifies that state ...

5 Some Basic Examples of Quantum Systems It is will be very useful for later studies to examine some of the mostfundamental quan ...

equation with〈n|, ih ̄ ∂ ∂t cn(t) =〈n|H|ψ(t)〉= ∑N m=1 〈n|H|m〉cm(t) (5.3) To obtain the second equality, we have inserted the ide ...

operatorT is simpler thanH, it will be advantageous to diagonalize it first. Since T is unitary, its eigenvalues are pure phases ...

5.1.3 The spectrum and generalized Hamiltonians We are guaranteed thatHis diagonalizable in this basis; in fact it is already di ...

5.2 Propagation in an infinite 1-dimensional lattice Keeping the lattice spacingafixed, we may let the physical extent of the la ...

This relation must hold for all states|k′;T〉, and it means that 〈k;T|k′;T〉= 2πδ(k′−k) k,k′∈[−kc,kc] (5.26) whereδ(k−k′) is theDi ...

The translation equationT†XT=X+aIthen allows us to compute the commutator [X,P] =i ̄hI (5.33) The position eigenstates become la ...

5.4 Propagation on the full line In the problem of propagation on the circle, we may finally take the infinite volume limit wher ...

5.4.1 The Diracδ-function The defining relation for the Diracδ-function on the real line is such that for all test functions f(a ...

5.5 General position and momentum operators and eigenstates Given self-adjoint position and momentum operators, denoted respecti ...

This choice of normalization uniquely determines the completeness relation, and we have^4 I = ∫ R dx|x;X〉〈x;X| I = ∫ R dk 2 π |k ...

so that|φ(k)|= 1. The phase ofφ(k) is not determined by the normalization conditions (5.58). The simplest choice is given byφ(k) ...

The solution to the harmonic oscillator problem will then provide an approximation to the problem for general potentialV for rea ...

Next, we show that the operatorH is bounded from below by^12 ̄hω. Indeed, take any normalized state|ψ〉, and compute the expectat ...

Since we have (see section 5.5 for the derivation), 〈x;X|X|ψ〉=xψ(x) 〈x;X|P|ψ〉=−i ̄h ∂ ∂x ψ(x) (5.80) The ground state wave funct ...

whereεijkis totally antisymmetric ini,j,kand is normalized toε 123 = 1. We now study all possible quantum systems on which the a ...

Given thatJ 1 andJ 2 are self-adjoint, the operatorsJ±are not self-adjoint, but are instead the adjoints of one another, (J+)†=J ...