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4.3: Normalization of the Wavefunction

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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. 1.2.2 , the probability of a measurement of x yielding a result between  and + is  is

Px:(t)=|ψ(x,t)|2dx

However, a measurement of x must yield a value between  and +, since the particle has to be located somewhere. It follows that Px:=1, or 

|ψ(x,t)|2dx=1

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 x=x0 and of characteristic width σ 

ψ(x)=ψ0e(xx0)2/(4σ2)

In order to determine the normalization constant  ψ0, we simply substitute Eq. 1.3.3 and into Eq. 1.3.2, to obtain

|ψ0|2e(xx0)2/(2σ2)dx=1

Changing the variable of integration to y=(xx0)/(2σ), we get

|ψ0|22σey2dy=1

However,
ey2dy=π

which implies that
|ψ0|2=1(2πσ2)1/2

Hence, a general normalized Gaussian wavefunction takes the form 

ψ(x)=eiφ(2πσ2)1/4e(xx0)2/(4σ2)

where φ is an arbitrary real phase-angle.

Now, it is important to demonstrate that if a wave function 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 wave function is untenable, since it does not make sense for the probability that a measurement of x yields any possible outcome (which is, manifestly, unity) to change in time. Hence, we require that

ddt|ψ(x,t)|2dx=0

for wavefunctions satisfying Schrödinger's equation. The above equation gives

ddtψψdx=(ψtψ+ψψt)dx=0

Now, multiplying Schrödinger's equation by ψ/(i), we obtain 

ψψt=i2mψ2ψx2iV|ψ|2

 

The complex conjugate of this expression yields
ψψt=i2mψ2ψx2+iV|ψ|2

[since (AB)=AB,A=A, and i=i]. Summing the previous two equations, we get

ψtψ+ψψt=i2m(ψ2ψx2ψ2ψx2)=i2mx(ψψxψψx)

Equations 1.3.10 and 1.3.13 can be combined to produce

ddt|ψ|2dx=i2m[ψψxψψx]=0

The above equation is satisfied provided
|ψ|0 as |x|

However, this is a necessary condition for the integral on the left-hand side of Eq. 1.3.2 to converge. Hence, we conclude that all wavefunctions which are square-integrable [i.e., are such that the integral in Eq. 1.3.2 converges] have the property that if the normalization condition (1.3.2) 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

dPxa:bdt+j(b,t)j(a,t)=0

where Pxa:b is defined in Eq. 1.2.2, and 

j(x,t)=i2m(ψψxψψx)

is known as the probability current. Note that j is real. Equation 1.3.16 is a probability conservation equation. According to this equation, the probability of a measurement of  x lying in the interval a to b evolves in time due to the difference between the flux of probability into the interval [i.e., j(a,t)], and that out of the interval [i.e., j(b,t)].  Here, we are interpreting j(x,t) as the flux of probability in the +x -direction at position x and time t. 

Note, finally, that not all wavefunctions can be normalized according to the scheme set out in Eq. 1.3.2. For instance, a plane wave wavefunction 

ψ(x,t)=ψ0ei(kxωt)

is not square-integrable, and, thus, cannot be normalized. For such wavefunctions, the best we can say is that

Pxa:b(t)ba|ψ(x,t)|2dx

In the following, all wavefunctions are assumed to be square-integrable and normalized, unless otherwise stated.

Contributors

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 4.3: Normalization of the Wavefunction is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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