Consider the electron wave function ψ(x)={ c √(1-x^2) ...

Consider the electron wave function $$\psi(x)=\left\{\begin{array}{ll}c \sqrt{1-x^{2}} & |x| \leq 1 \mathrm{cm} \\0 & |x| \geq 1 \mathrm{cm}\end{array}\right.$$ where $x$ is in $\mathrm{cm}$. a. Determine the normalization constant $c$ b. Draw a graph of $\psi(x)$ over the interval $-2 \mathrm{cm} \leq x \leq 2 \mathrm{cm} .$ Provide numerical scales on both axes. c. Draw a graph of $|\psi(x)|^{2}$ over the interval $-2 \mathrm{cm} \leq x \leq$ 2 cm. Provide numerical scales. d. If $10^{4}$ electrons are detected, how many will be in the interval $0.00 \mathrm{cm} \leq x \leq 0.50 \mathrm{cm} ?$

Consider the electron wave function $$\psi(x)=\left\{\begin{array}{ll}c \sqrt{1-x^{2}} & |x| \leq 1 \mathrm{cm} \\0 & |x| \geq 1 \mathrm{cm}\end{array}\right.$$
where $x$ is in $\mathrm{cm}$.
a. Determine the normalization constant $c$
b. Draw a graph of $\psi(x)$ over the interval $-2 \mathrm{cm} \leq x \leq 2 \mathrm{cm} .$ Provide numerical scales on both axes.
c. Draw a graph of $|\psi(x)|^{2}$ over the interval $-2 \mathrm{cm} \leq x \leq$
2 cm. Provide numerical scales.
d. If $10^{4}$ electrons are detected, how many will be in the interval
$0.00 \mathrm{cm} \leq x \leq 0.50 \mathrm{cm} ?$

Key Concepts

Wavefunction Normalization

Normalization is a fundamental concept in quantum mechanics requiring that the total probability of finding a particle over all space be equal to one. This involves determining a constant such that the integral of the modulus squared of the wavefunction over its domain is unity. It ensures the wavefunction properly represents a probability amplitude.

Probability Density

The probability density is given by the modulus squared of the wavefunction. It provides the likelihood of finding a particle at a specific position. In quantum mechanics, interpreting |?(x)|² as a probability per unit length (or volume) is crucial for making physical predictions about measurement outcomes.

Probability Calculation Using Integration

Calculating the probability of finding a particle within a specific interval involves integrating the probability density over that interval. This process allows one to determine the relative number of particles expected in different regions, making it an essential method for connecting theoretical predictions with experimental measurements.

Graphical Representation of Quantum States

Visual representations, such as graphs of the wavefunction and its probability density, are important for understanding the spatial behavior of quantum systems. Graphs provide intuitive insights into features like symmetry, regions of high probability, and the effects of normalization, aiding in the interpretation of the underlying physics.

Transcript

00:01 To determine the normalization constant, we simply use the normalization condition.

00:07 So, for a, we have that size squared, integrated around x.

00:23 This is, of course, size of x.

00:27 This needs to be one.

00:32 Since the way function is only non -zero, if it is one centimeter or less from the origin, then we need to integrate from minus 1 to 1 cm and then this gives c squared right and then we have x minus x to 3 over 3 this is from minus 1 to 1 since this is 1 and this is 1 and this whole thing is simply 4 thirds we have we have that c needs to be, so we'll take the positive value, square root of three -fourths, which is around 0 .866 inverse square root centimeters.

01:48 So that is c.

01:52 For b, we just need to draw a graph of si.

01:58 So, okay, we know that si is that si is a square root of 1 minus.

02:16 X squared.

02:18 So at 0 it's just c which is square root of three fourths and we know that for example at minus 1 and 1 it's it needs to be 0.

02:38 So we'll put it here and here.

02:45 I'll draw it in blue.

02:47 So it's 0 here.

02:54 Now for example at 0 .5 it is c times square.

03:00 Square root of 1 minus 1 fourth.

03:06 So it's square root of 3 fourths.

03:10 There is also c, so it's c squared.