Q2) At a particular time of interest, the state function for...

Q2) At a particular time of interest, the state function for a particle of mass m is of the form $\psi(x) = Nx^{\frac{1}{2}}$ with $\psi(x) = 0$ elsewhere. Answer the following • assuming the above $\psi(x)$ for all parts. • using the value of N you obtain in part (a) for parts (b)-(d), evaluating as far as possible without knowing L. • evaluating all integrals in all parts as far as possible. (a) Evaluate the normalization constant N. (b) What is the probability that position measurement would find the particle within $dx$ of the position $x = \frac{L}{2}$? (c) Evaluate the probability that position measurement would find the particle to be in the region $0 \leq x \leq \frac{L}{2}$. (d) Evaluate the expected value of the particle position $x$.

Transcript

00:01 Okay, so in this problem we have a wave function and we must calculate a lot of things in this problem.

00:07 So first of all, in part a, we have to find the normalization constant of the wave function.

00:15 And to find the normalization constant, we must make an integral in the interval that goes from zero to infinity of the psi, which is the wave function.

00:31 That multiplies deep -si star, which is the complex conjugated of this wave function.

00:38 Integrated in the x equals 1.

00:42 So that's the definition of normalization.

00:45 And we must do this integral to find the normalization constant.

00:51 Now in part b, in part b, let's see what we want to calculate.

00:56 We have to find the probability that a particle can be found on the interval, from 0 to l.

01:05 Well to calculate the probability again we just need to make the integral from 0 to l of psi psi star the x okay so that's what we must calculate in the part b now for part c in part c we have to find the expectation value of position.

01:37 So to find expectation value of position, we must do integral from minus infinity to infinity, but if we look to this function, we know that this function is defined from 0 to l.

01:59 So if the function is defined from 0 to l, instead we use the negative infinity and the positive infinity let's just use 0 to l so this is 0 to l and the average of the position actually let's call this the expectation value of position can be described this quantity here which is the integral from 0 to l in this particular wave function from psi x, psi star, d x.

02:46 So that's what we must calculate in part c.

02:49 Now for part d.

02:54 In part d, we must find the expectation value of the connecting energy.

03:01 So again, let's put here an e, e, k, which is going to be the expectation value of the connecting energy, which, which is going to be just the integral from 0 to l again of psi, the operator of the energy, psi, star, the x.

03:31 So here is what we must calculate in each one of the parts of this problem.

03:38 So we can now begin with part a, which is the normalization.

03:44 As we can see here, we just need to make this integral.

03:48 And this is going to be, let's see, from part a, we have that.

03:59 This is going to be the integral from 0 to infinity of a square, x square, the e minus 2 alpha x.

04:20 Dx.

04:22 Just an observation the complex conjugated of the function is when we change negative is when we change the imaginary number for negative imaginary number.

04:36 So it's just this we go from e i for minus i.

04:42 Okay, this is take the conjugate of a function...

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