# What is the Most Likely Position of a Particle in 1-Dimensional Wavefunction?

In summary, the conversation discusses finding the most likely position of a particle using the equation \Psi = A[(x+1)^{2} - 1)] and the expected value of the particle's position, <x>. The maximum of psi*conjugate(psi) is the expected value of the particle's position and can be found by taking the derivative of |psi*conjugate(psi)| and setting it to 0. The maximum of psi*psi is the same as the maximum of |psi|.

## Homework Statement

Find the most likely position of the particle.

## Homework Equations

$$\Psi$$ = A[(x+1)$$^{2}$$ - 1)]
between x = 0 and x = 1
$$\Psi$$ = 0 anywhere else

## The Attempt at a Solution

I found A to equal $$\sqrt{15 / 38}$$... but I am not sure how to do the rest of it

You didn't even have to do that if you just want the most likely position. That's at the maximum of psi*conjugate(psi), the probability density.

that gives you expected value of the particles position <x>, which is the next question,
i need to find the most likely position of the particle...
is there another way to do this?

that gives you expected value of the particles position <x>, which is the next question,
i need to find the most likely position of the particle...
is there another way to do this?

No, <x> is the integral of x*psi*conjugate(psi) over the integral of psi*conjugate(psi). The 'most likely position' is the maximum of psi*conjugate(psi).

so take the derivative of |psi*cojugate(psi)| and set it to 0?

the conjugate would be A[x+1)^2 +1] ?

so take the derivative of |psi*cojugate(psi)| and set it to 0?

the conjugate would be A[x+1)^2 +1] ?

Yep. And since psi is real, conjugate(psi)=psi. You'll probably notice that the maximum of psi*psi is the same as the maximum of |psi|.

thanks a bunch =)

## 1. What is a wavefunction in 1-dimension?

A wavefunction in 1-dimension is a mathematical representation of the probability amplitude for a particle to be found at a specific position along a one-dimensional axis. It describes the quantum state of a particle in terms of its position, momentum, and energy.

## 2. How is a wavefunction in 1-dimension used in quantum mechanics?

In quantum mechanics, the wavefunction in 1-dimension is used to describe the behavior of particles on a one-dimensional axis. It is a fundamental concept that helps to predict the probabilities of different outcomes of a measurement, such as the position or momentum of a particle.

## 3. What does the wavefunction in 1-dimension tell us about a particle?

The wavefunction in 1-dimension provides information about the position, momentum, and energy of a particle in a one-dimensional space. It also gives us insights into the probability of finding the particle at a particular location or with a specific momentum.

## 4. How is the wavefunction in 1-dimension calculated?

The wavefunction in 1-dimension is calculated using the Schrödinger equation, which is a mathematical formula that describes the evolution of a quantum system over time. It takes into account the potential energy of the system and the initial conditions to determine the wavefunction at any given point in time.

## 5. Can the wavefunction in 1-dimension change over time?

Yes, the wavefunction in 1-dimension can change over time according to the Schrödinger equation. As the quantum system evolves, the wavefunction will change, and the probabilities of different outcomes will also change. This is one of the key principles of quantum mechanics known as wavefunction collapse.

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