A particle is described by the wave function ψ(x)={ A cos((2 π x)/(L)) for -(L)/(4)≤ x ≤

A particle is described by the wave function ψ(x)={ A...

A particle is described by the wave function $$\psi(x)=\left\{\begin{array}{ll}A \cos \left(\frac{2 \pi x}{L}\right) & \text { for }-\frac{L}{4} \leq x \leq \frac{L}{4} \\0 & \text { otherwise }\end{array}\right.$$ (a) Determine the normalization constant $A$. (b) What is the probability that the particle will be found between $x=0$ and $x=L / 8$ if a measurement of its position is made?

A particle is described by the wave function
$$\psi(x)=\left\{\begin{array}{ll}A \cos \left(\frac{2 \pi x}{L}\right) & \text { for }-\frac{L}{4} \leq x \leq \frac{L}{4} \\0 & \text { otherwise }\end{array}\right.$$
(a) Determine the normalization constant $A$. (b) What is the probability that the particle will be found between $x=0$ and $x=L / 8$ if a measurement of its position is made?

Key Concepts

Wave Function Normalization

In quantum mechanics, every valid state of a system is described by a wave function that must be normalized so that the total probability of finding the particle somewhere in space is equal to one. This is achieved by integrating the absolute square of the wave function over all space and setting the integral equal to one, which allows for the determination of any normalization constant present in the function.

Probability Density and Measurement

The square of the wave function, |?(x)|², represents the probability density for finding a particle at a given position. When a measurement is made, the probability of locating the particle in a specific region is obtained by integrating the probability density over that region. This connection between the wave function and observable probabilities is a cornerstone of the quantum mechanical description of nature.

Integration Techniques in Quantum Mechanics

Integration is a fundamental mathematical tool in quantum mechanics used to normalize wave functions and calculate probabilities. By performing definite integrals of |?(x)|² over appropriate intervals, one can confirm that the state is properly normalized and determine the likelihood of finding the particle in particular segments of space.

Transcript

00:01 We start by calculating the integral from minus l over 4 to l over 4 of cosine squared 2 pi x over l, dx equal to 1, so we can get a.

00:30 And we're going to scroll down a bit to make some room.

00:33 So we want this integral, first of all.

01:10 This is equal to x over 2 plus l over 8 pi, sign 4 pi x over l evaluated from minus l over 4 to l over 4.

01:32 I get a lot of times i get these types of integrals from tables but you can also use tricks like integration by parts to evaluate these integrals but for the purposes of these videos it's more convenient to just write the integral down as a result so proceeding, we go ahead and we evaluate at the endpoints, and we have a squared, one -half, l over 4, minus negative l over 4.

02:14 And then we look at the sine function.

02:18 We notice we have 4 pi times x.

02:21 If i substitute l over 4 or minus l over 4, either way, that's supposed to be, there's not supposed to be that.

02:31 Stray mark there.

02:37 So that was meant to be l over 4 to indicate that if we were to place that in x and multiply 4 pi times l over 4 and divide by l, i would simply have pi.

02:51 So you would end up with sine of pi or sign of minus pi.

02:59 Either way, you would have zero.

03:02 So we get no contribution from sign, from either of these terms at the endpoints.

03:11 So we have 0 minus 0, and we're left with only these contributions from l over 4 and negative l over 4.

03:19 These two minus signs, by the way, will multiply to give us a positive sign.

03:24 So we now have l over 2 because l over 4 plus l over 4 gives us l over 2...

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