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Now, a probability is a real number between 0 and 1. An outcome
of a measurement which has a probability 0 is an impossible outcome, whereas an
outcome which has a probability 1 is a certain outcome. According to
Eq. (138), the probability of a measurement of yielding
a result between and is

(139) 
However, a measurement of must yield a value between and
, since the particle has to be located somewhere. It follows that
, or

(140) 
which is generally known as the normalization condition for the
wavefunction.
For example, suppose that we wish to normalize the wavefunction of
a Gaussian wave packet, centered on , and of characteristic
width (see Sect. 3.12): i.e.,

(141) 
In order to determine the normalization constant , we simply substitute
Eq. (141) into Eq. (140), to obtain

(142) 
Changing the variable of integration to
, we get

(143) 
However,

(144) 
which implies that

(145) 
Hence, a general normalized Gaussian wavefunction takes the form

(146) 
where is an arbitrary real phaseangle.
Now, it is important to demonstrate that if a wavefunction is initially
normalized then it stays normalized as it evolves in time according
to Schrödinger's equation. If this is not the case then
the probability interpretation of the wavefunction is untenable, since it
does not make sense for the probability that a measurement of yields any possible outcome (which is, manifestly, unity) to change in time.
Hence, we require that

(147) 
for wavefunctions satisfying Schrödinger's equation.
The above equation gives

(148) 
Now, multiplying Schrödinger's equation by
,
we obtain

(149) 
The complex conjugate of this expression yields

(150) 
[since
,
, and ].
Summing the previous two equations, we get

(151) 
Equations (148) and (151) can be combined to
produce

(152) 
The above equation is satisfied provided

(153) 
However, this is a necessary condition for the integral on the lefthand
side of Eq. (140) to converge. Hence, we conclude that
all wavefunctions which are squareintegrable [i.e., are such that the integral in Eq. (140) converges] have the property
that if the normalization condition (140) is satisfied at one instant
in time then it is satisfied at all subsequent times.
It is also possible to demonstrate, via very similar analysis to the above, that

(154) 
where
is defined in Eq. (138), and

(155) 
is known as the probability current. Note that is real.
Equation (154) is a probability conservation equation.
According to this equation, the probability of a measurement
of lying in the interval to evolves in time due to the
difference between the flux of probability into the interval [i.e., ],
and that out of the interval [i.e., ].
Here, we are interpreting as the flux of probability in the direction at position and time .
Note, finally, that not all wavefunctions can be normalized according to the scheme set out in Eq. (140). For instance, a plane wave wavefunction

(156) 
is not squareintegrable, and, thus, cannot be normalized.
For such wavefunctions, the best we can say is that

(157) 
In the following, all wavefunctions are assumed to be squareintegrable and
normalized, unless otherwise stated.
Next: Expectation Values and Variances
Up: Fundamentals of Quantum Mechanics
Previous: Schrödinger's Equation
Richard Fitzpatrick
20100720