Consider a particle moving in one dimension which we shall...

Consider a particle moving in one dimension, which we shall call the $x$-axis. (a) What does it mean for the wave function of this particle to be $normalized$? (b) Is the wave function $\psi(x) = e^{ax}$ , where a is a positive real number, normalized? Could this be a valid wave function? (c) If the particle described by the wave function $\psi(x) = Ae^{-bx}$, where $A$ and $b$ are positive real numbers, is confined to the range $x \geq 0$, determine A (including its units) so that the wave function is normalized.

Transcript

00:01 In question a, we have to discuss what it means for the wave function to be normalized.

00:05 So consider that we have a certain wave function phi of x.

00:10 And remember that the probability density that's associated with this function is the modulus of phi of x squared.

00:22 This is the probability density, so i'm going to just put a d index here.

00:26 And the probability that we find the particle between a and b, x, a is smaller than x, which is smaller than b, is the integral from a to b of the probability density, dx.

00:54 So the probability that we found the particle somewhere that is between in x, which is between minus infinity and infinity, it's equal to the integral from minus infinity to infinity of the modules of phi x squared dx.

01:23 But notice that the particle must be somewhere.

01:26 So this means that the probability must be 1.

01:32 And this is what the normalization of the wave function means.

01:37 Well, basically, we are imposing, when we normalize the wave function, we impose that this quantity here is equal to 1.

01:46 But as i said, this quantity means that the probability of finding the particle somewhere is 1.

01:55 That is, the particle must be somewhere.

01:59 So basically, the normalization of the wave function means that the particle must be somewhere.

02:19 Then in question b, we have the wave function e to the ax, where a is a positive constant, and we have to decide whether or not this function is normalized, and whether or not it's even a possible wave function...

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