# Calculate |Psi(x,t)|^2 - Merzbacher Chapter 15 Exercise 9

• IHateMayonnaise
In summary, the conversation involves a question about Exercise 9 from chapter 15 of the book "merzbacher," which asks to find the value of \lvert\psi(x,t)\rvert^2. The equations involved include \lvert\psi(x,t)\rvert^2=\psi^*(x,t)\psi(x,t), (\Delta x)^2=\langle x \rangle^2 - \langle x^2\rangle, and i\hbar\frac{d}{dt}\langle A \rangle = \langle[A,H]\rangle + \left\langle\frac{\partial A}{\partial t} \right\rangle. The person asking the question is wondering if there is an easier way to
IHateMayonnaise

## Homework Statement

This is Exercise 9 from chapter 15 of merzbacer. It asks to find $\lvert\psi(x,t)\rvert^2$ given:

$$\psi(x,t)=[2\pi(\Delta x)_0^2]^{-1/4}\left[1+\frac{i\hbar t}{2m(\Delta x)_0^2} \right]^{-1/2} \exp\left[\frac{-\frac{x^2}{4(\Delta x)_0^2}+ik_0x-ik_0^2\frac{\hbar t}{2m}}{1+\frac{i\hbar t}{2m(\Delta x)_0^2}}\right]$$

## Homework Equations

$$\lvert\psi(x,t)\rvert^2=\psi^*(x,t)\psi(x,t)$$

$$(\Delta x)^2=\langle x \rangle^2 - \langle x^2\rangle$$

$$i\hbar\frac{d}{dt}\langle A \rangle = \langle[A,H]\rangle + \left\langle\frac{\partial A}{\partial t} \right\rangle$$

## The Attempt at a Solution

My question is quick and qualitative: is there..an easier way of doing this than the brute force way? I mean, am I not seeing something? Or is this problem as useless as it seems?

I am not allowed to use a computer in any way.

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It looks like lots of stuff will cancel out when you do it "brute force." But, you're not looking for any particular final value (e.g. the expectation value of 'x,' etc), so there isn't really anything to take a short-cut towards.

Remember that in order to calculate the integral

$$\exp[i\theta]\cdot \exp[-i\theta]=1$$

where theta is the content of the exponential part
So calculations reduce.

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## 1. What is the purpose of calculating |Psi(x,t)|^2 in Merzbacher Chapter 15 Exercise 9?

The purpose of this exercise is to determine the probability density of finding a particle at a specific position and time, based on the wave function Psi(x,t).

## 2. What is the formula for calculating |Psi(x,t)|^2?

The formula is |Psi(x,t)|^2 = Psi(x,t) * Psi*(x,t), where Psi*(x,t) is the complex conjugate of Psi(x,t).

## 3. How is the probability density related to the wave function?

The probability density is directly proportional to the square of the wave function. This means that as the wave function increases or decreases, the probability density will also increase or decrease accordingly.

## 4. Can the probability density be negative?

No, the probability density must always be a positive value. This is because it represents the likelihood of finding a particle at a specific position and time, which cannot have a negative probability.

## 5. How can we interpret the results of calculating |Psi(x,t)|^2?

The results represent a probability distribution, showing the relative likelihood of finding a particle at different positions and times. The highest peaks in the distribution correspond to the most probable locations for the particle to be found.

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