14.6.5 Summary of the structure of quantum mechanics
We can now summarize the logical structure of quantum me-
chanics using the following five principles.- Wavefunction fundamentalism:All knowable information about
a system is encoded in its wavefunction (ignoring phase and
normalization). - Inner product: Wavefunctions come equipped with an inner
product that has the properties〈u|αv+βw〉=α〈u|v〉+β〈u|w〉
and〈u|v〉=〈v|u〉∗. - Observables:In quantum mechanics, any observable is repre-
sented by a linear operatorOthat takes a wavefunction as an
input and is hermitian,〈Ou|v〉=〈u|Ov〉. - Unitary evolution of the wavefunction:The wavefunction evolves
over time, according to the Schr ̈odinger equationi~∂Ψ/∂t=
HˆΨ, in a deterministic manner. BecauseHˆ is an observable,
the Schr ̈odinger equation islinearand alsounitary. Unitarity
means that〈u(t)|v(t)〉 =〈u(t′)|v(t′)〉, so that probability is
conserved and information is never lost. - Completeness:For any system of interest, there exists a set of
compatible observables, called a complete set, such that any
state of the system can be expressed as a sum of wavefunctions
having definite values of these observables.
14.7 Applications to the two-state system
14.7.1 A proton in a magnetic field
As an application of the ideas discussed in section 14.6, let us
consider the example of a proton at rest in a uniform magnetic field.
We will find that this very simple example has surprising properties,
and also that it throws light on much more general ideas than would
be expected, given how specific the situation is. We discuss the
proton because the physics is then the physics of nuclear magnetic
resonance (NMR), which is the technology used for, among other
things, medical MRI scans.
Classically, the proton feels no magnetic force because it is at
rest, and also because the field is uniform (unlike the one in the
Stern-Gerlach experiment). Therefore we expect it to stay at rest.
Its energy is−m·B, and for the reasons discussed in sec. 11.2.4,
p. 695, the magnetic dipole momentmis proportional to the spin
angular momentum vectors, so that the energy can be broken up
into a sum of three terms asksxBx+ksyBy+kszBz, wherekis
− 1 /gtimes the proton’s charge-to-mass ratio.990 Chapter 14 Additional Topics in Quantum Physics