A particle is described in 1-D by a wavefunction: ψ=Be^-2 x for x ≥0 and Ce^+4 x for x<0, and B

A particle is described in 1-D by a wavefunction: ψ=Be^-2 x...

A particle is described in 1-D by a wavefunction: $\psi=\mathrm{Be}^{-2 x}$ for $x \geq 0$ and $\mathrm{Ce}^{+4 x}$ for $x<0$, and $\mathrm{B}$ and $\mathrm{C}$ are real constants. Calculate $\mathrm{B}$ and $\mathrm{C}$ to make $\psi$ a valid wavefunction. Where is the particle most likely to be?

A particle is described in 1-D by a wavefunction:
$\psi=\mathrm{Be}^{-2 x}$ for $x \geq 0$ and $\mathrm{Ce}^{+4 x}$ for $x<0$, and $\mathrm{B}$ and $\mathrm{C}$ are real constants. Calculate $\mathrm{B}$ and $\mathrm{C}$ to make $\psi$ a valid wavefunction. Where is the particle most likely to be?

Key Concepts

Wavefunction Normalization

In quantum mechanics, a wavefunction must be normalized so that the total probability of finding the particle anywhere in space is 1. This involves integrating the square of the absolute value of the wavefunction over all space and setting that integral equal to 1. In problems with piecewise-defined functions, the integration is carried out separately over each region and their contributions are summed.

Continuity of the Wavefunction

A physical wavefunction must be continuous (and often its derivative must also be continuous) at boundaries where the potential is finite. This requirement ensures that the probability density is well-behaved and there are no abrupt, unphysical jumps in the function. For a piecewise-defined wavefunction, the values at the boundary of the pieces are set equal to each other.

Probability Density and Most Likely Location

The probability density is given by the square of the magnitude of the wavefunction. To determine the most likely location of the particle, one examines where this probability density is maximized. In problems like the given one, the maximum probability density generally occurs at the point where the wavefunction reaches its highest absolute value after normalization and is continuous.

Transcript

00:01 Here we're given a wave function and we're asked to do various things to it.

00:04 So the first one just asks us to find and sketch the probability density.

00:10 So probability density is just the way function squared.

00:15 So this will be equal to this.

00:18 It's going to be 2 over a, e to the negative 2 x over a or x is greater than 0.

00:27 And then when x is less than 0, this is 0.

00:30 So let's sketch this out.

00:38 So just meet the x -axis and then here is si squared.

00:43 Now at x -equals 0, what we see is that this is going to be equal to 2 over a.

00:49 So this is 2 over a.

00:52 Now on the negative axis, this is just 0.

00:55 So there's no probability of the electron being over there.

00:59 But when x is greater than 0, this ends up.

01:04 Decane exponentially.

01:06 So it's going to look something like this as this function goes to zero as x goes to infinity.

01:14 Now part b has to find the probability that the particle will be anywhere, anywhere with x is less than zero.

01:26 So to do that, we would have to do this integral of the probability distribution, from negative infinity to zero since that's where they ask us about, right? however, of course, this is simply just going to be the integral of zero itself, which is zero, right? i mean, we can easily see that's here.

01:50 The probability of finding a particle at some place is just the area underneath this curb, but here there is no curve at all, right? everything's equal to zero, and so the area is equal to zero as well.

02:06 Okay, now part c asks us to show that this is normalized the way function.

02:15 And so all we have to do is calculate the integral from zero to infinity out of psi squared.

02:21 And we don't do negative infinity because the way function is zero there...

Please give Ace some feedback