String Theory Demystified

4 String Theory Demystifi ed
detail, we will review the basic ideas of quantum mechanics in this section (see
Quantum Mechanics Demystifi ed for a detailed description). In quantum mechanics,
everything we could possibly fi nd out about a particle is contained in the state of the
particle or system described by a wave function:
ψ ( x, t )
The wave function is a solution of Schrödinger’s equation:
2
2
∂ψ
− ∇ ψ + Vψ = i
2m
∂t
(1.5)
The wave function itself is not a real physical wave, rather it is a probability
2
amplitude whose modulus squared ψ ( x, t)
(note that the wave function can be
complex) gives the probability that the particle or system is found in a given
state.
Measurable observables like position and momentum are promoted to
mathematical operators in quantum mechanics. They act on states (i.e., on wave
functions) and must satisfy certain commutation rules. For example, position and
momentum satisfy
[ x, p] = i
(1.6)
Furthermore, there exists an uncertainty principle that puts constraints on the
precision with which certain quantities can be known. Two important examples are
∆x ∆p≥/ 2
∆E ∆t ≥/
2
(1.7)
So the more precisely we know the momentum of a particle, the less certain we are
of its position and vice versa. The smaller the interval of time over which we
examine a physical process, the greater the fl uctuations in energy.
When considering a system with multiple particles, we have a wave function
ψ ( x1, x2, … , xn) say where there are n particles with coordinates xi . It turns out
that there are two basic types of particles depending on how the wave function
behaves under particle interchange xi xj.
Considering the two-particle case for
simplicity, if the sign of the wave function is unchanged under
ψ( x , x ) =
ψ(
x , x )
1 2 2 1