QuantumPhysics.dvi

polarization angleθ=α= 0, while the state|y〉corresponds toθ =α=π/2. From this correspondence, we deduce the probabilities, and p ...

The instruments of the polarizer, analyzer (which, recall, is just a polarizer) and the bire- fringent plate have mathematical i ...

Figure 4: The Stern-Gerlach apparatus SG deals with measuringS, or equivalentlym. To measurem, we place it in a magnetic fieldB, ...

(1)With a strong magnetic field, the signal of the individual atoms emerging from SG is centered sharply on two outcomes, which ...

probability spinsSx=± ̄h/2. The third set-up shows how successive SG apparatus inz,x and then again inzdirections regenerate spi ...

3 Mathematical Formalism of Quantum Physics We begin by reviewing the mathematical formalism of quantum physics, including Hilbe ...

• The norm onHprovides with a natural notion of distance, which induced the distance topology onH, and allows us to investigate ...

ofN(the set of positive integers). The Gramm-Schmidt procedure goes as follows. We start with the first vector and normalize it, ...

3.1.4 Finite-dimensional Hilbert spaces All Hilbert spaces of finite dimensionNare isomorphic to one another. Let{|n〉}n=1,···,Nb ...

A more complicated example is provided by spaces of square integrable complex functions on some intervalS(or all of) the real li ...

3.2 Linear operators on Hilbert space A linear operator (or simply operator)AonHis a linear map fromHtoH. Its action on a state| ...

Clearly,Dhas a well-defined action on every basis vector. But now consider itsaction on a general vector|φ〉 ∈ H, with the decomp ...

a non-trivial subspace E ⊂ H, which is itself a Hilbert space. The associated projection operatorPEmay be constructed uniquely i ...

for all|φ〉,|ψ〉∈H, while a self-adjoint bounded operator satisfies〈ψ|A|φ〉=〈φ|A|ψ〉∗for all |φ〉,|ψ〉∈H. A projection operator is alw ...

(ii) Next, leta′ 6 =abe two distinct eigenvalues (which are both real by (i)), A|φ〉 = a|φ〉 A|φ′〉 = a′|φ′〉 (3.40) Taking the inne ...

3.4.1 Unitary operators Unitary operators are very closely related to self-adjoint operators. In fact, one proves the following ...

is a unitary matrix. The statement is readily checked by computingU† =e−iA † =e−iA. Conversely, any unitary matrixUmay be writte ...

4 The Principles of Quantum Physics The mathematical description of a physical quantum system is in terms of a separable Hilbert ...

4.1 Conservation of probability The combination of Principles 2 and 3 specifies the nature of measurement of a general observabl ...

compatible observables, [px,py] = 0, while the spin componentsSxandSyare not, [Sx,Sy] = i ̄hSz. The role played by compatibility ...