Page 95 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

86 General principles of quantum theory

State function
According to the ®rst postulate, the state of a physical system is completely
described by a state function Ø(q, t) or ket jØi, which depends on spatial
coordinates q and the time t. This function is sometimes also called a state
vector or a wave function. The coordinate vector q has components q 1 , q 2 , ... ,
so that the state function may also be written as Ø(q 1 , q 2 , ... , t). For a particle
or system that moves in only one dimension (say along the x-axis), the vector q
has only one component and the state vector Ø is a function of x and
t: Ø(x, t). For a particle or system in three dimensions, the components of q
are x, y, z and Ø is a function of the position vector r and t: Ø(r, t). The state
function is single-valued, a continuous function of each of its variables, and
square or quadratically integrable.

For a one-dimensional system, the quantity Ø (x, t)Ø(x, t) is the probabil-
ity density for ®nding the system at position x at time t. In three dimensions,

the quantity Ø (r, t)Ø(r, t) is the probability density for ®nding the system at

point r at time t. For a multi-variable system, the product Ø (q 1 , q 2 ,
... , t)Ø(q 1 , q 2 , ... , t) is the probability density that the system has coordi-
nates q 1 , q 2 , ... at time t. We show below that this interpretation of Ø Ø

follows from postulate 3. We usually assume that the state function is normal-
ized

Ø (q 1 , q 2 , ... , t)Ø(q 1 , q 2 , ... , t)w(q 1 , q 2 , ...)dq 1 dq 2 ... ˆ 1
or in Dirac notation
hØjØiˆ 1
where the limits of integration are over all allowed values of q 1 , q 2 , ...

Physical quantities or observables
The second postulate states that a physical quantity or observable is represented
in quantum mechanics by a hermitian operator. To every classically de®ned
function A(r, p) of position and momentum there corresponds a quantum-
^
mechanical linear hermitian operator A(r,("=i)=). Thus, to obtain the quan-
tum-mechanical operator, the momentum p in the classical function is replaced
by the operator ^ p
"
^ p ˆ = (3:44)
i
or, in terms of components
" @ " @ " @
^ p x ˆ , ^ p y ˆ , ^ p z ˆ
i @x i @ y i @z