Shows the wave function of a particle confined between...

shows the wave function of a particle confined between $x=-4.0 \mathrm{mm}$ and $x=4.0 \mathrm{mm} .$ The wave function is zero outside this region. a. Determine the value of the constant $c,$ as defined in the figure. b. Draw a graph of the probability density $P(x)=|\psi(x)|^{2}.$ c. Draw a dot picture showing where the first 40 or 50 particles might be found. d. Calculate the probability of finding the particle in the interval $-2.0 \mathrm{mm} \leq x \leq 2.0 \mathrm{mm}.$ (FIGURE CANNOT COPY)

shows the wave function of a particle confined between $x=-4.0 \mathrm{mm}$ and $x=4.0 \mathrm{mm} .$ The wave function is zero outside this region.
a. Determine the value of the constant $c,$ as defined in the figure.
b. Draw a graph of the probability density $P(x)=|\psi(x)|^{2}.$
c. Draw a dot picture showing where the first 40 or 50 particles might be found.
d. Calculate the probability of finding the particle in the interval $-2.0 \mathrm{mm} \leq x \leq 2.0 \mathrm{mm}.$
(FIGURE CANNOT COPY)

Key Concepts

Integration to Compute Specific Probabilities

Calculating the probability of finding a particle within a certain interval (like between x = -2.0 mm and x = 2.0 mm) involves integrating the probability density |?(x)|² over that interval. This integral gives the total probability of encountering the particle in that specific spatial region.

Quantum Confinement in a Potential Well

The situation described is a classic example of quantum confinement, where a particle is restricted to a finite region of space (here, between x = -4.0 mm and x = 4.0 mm) with the wave function dropping to zero outside that region. This confinement leads to quantized states and discrete energy levels within the potential well.

Probabilistic Interpretation and Measurement in Quantum Mechanics

In quantum mechanics, the outcomes of measurements are interpreted probabilistically, meaning that even if the wave function provides a distribution, individual measurements yield specific, unpredictable results. The dot picture is a visual representation of many measurements where the spread of points corresponds to the probability density.

Normalization of the Wave Function

In quantum mechanics, the wave function must be normalized so its total probability over all space is 1. This involves determining a constant (here labeled c) such that the integral of |?(x)|² over the region where the particle exists equals unity. This concept ensures that the prediction of finding the particle somewhere in space is certain.

Probability Density

The probability density is given by |?(x)|², which provides a measure of the likelihood of finding the particle at a given position. Graphing this function or representing it with a dot picture helps predict where a particle is most likely to be observed when measurements are made.

Transcript

00:01 So we are given a wave function, the graph of a wave function, which just as a linear branch and then a null branch, zero branch.

00:14 So it's linear between minus four and four and then null zero otherwise.

00:20 And to determine the value of the constant c, we need to take into account this normalization condition.

00:28 So this means that the integral, so what's the integral? the integral is just the area under this graph.

00:39 And we know that, well, we know that this, the square of the absolute value of si, since si, so this, si is real, so we don't care about the absolute value.

00:59 But si is simply from minus 4 to 4 is c times x.

01:06 Or should i say that it's actually c over 4 times x so we just need to find the area beneath this this is simply c over 4 times x everything squared the x from minus 4 to 4 x squared so every this can be put out of the integral so it's just integrating x squared from minus 4 to 4 or 2 times from 0 to 4 if you will.

02:01 And this will simply give 8 thirds of c squared.

02:10 So this means that c is equal to the square root of 3 8s.

02:17 And of course, this has the units of inverse square root millimeters.

02:26 So that's c.

02:28 Now the second question asks us to draw the probability density, which is the same as, well, this.

02:40 Right? so here, the probability density is always positive.

02:53 And it's simply c squared over 16 times x squared, right? so i can write here c squared over 16 times x squared, with c squared being 3 over 8.

03:12 So the graph for this will simply be a quadratic function.

03:23 It's a little ugly.

03:26 So something like this.

03:28 And then goes to zero.

03:35 And this is minus four.