If the wave function is normalized, what is the probability density at x?

In summary, the wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value 3x^2/L^3.
  • #1
hidemi
208
36
Homework Statement
How to know the answer is D``?
Relevant Equations
non
The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value?

(A) 2x/L^2. (B) 2x^2 / L^2. (C) 2x^2 /L^3. (D) 3x^2 / L^3. (E) 3x^3 / L^3

Ans : D
 

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  • #2
Is there a question here ?
 
  • #3
I see you are pretty new here, so
:welcome: !​

And the PF guidelines ask that you post your own attempt at solution. What did you find ?
 
  • #4
hidemi said:
Homework Statement:: How to know the answer is D``?
Relevant Equations:: non

The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value?

(A) 2x/L^2. (B) 2x^2 / L^2. (C) 2x^2 /L^3. (D) 3x^2 / L^3. (E) 3x^3 / L^3

Ans : D
Let me help some more: the homework statement is the part in italics.
Your question (which I missed :rolleyes: ) is the part in red

And the answer follows from calculating A and then the probability density. For that you need the relevant equations. 'non' doesn't do it.

Hint: do the normalizatiion.

##\ ##
 
  • #5
BvU said:
I see you are pretty new here, so
:welcome: !​

And the PF guidelines ask that you post your own attempt at solution. What did you find ?
I think I got it, thanks!
 
  • Like
Likes BvU

What does it mean for a wave function to be normalized?

A wave function is said to be normalized if its total probability is equal to 1. This means that the chances of finding the particle described by the wave function in any location is 100%.

What is the mathematical relationship between the wave function and the probability density?

The probability density at a specific point x is equal to the absolute value squared of the wave function at that point, multiplied by a normalization constant. This can be expressed as P(x) = |Ψ(x)|^2.

How is the probability density related to the uncertainty principle?

The probability density is directly related to the uncertainty principle, as it represents the likelihood of finding a particle at a specific location. The uncertainty principle states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa. Therefore, the probability density at a given point x can never be exactly 0, but it can be very small.

Can the probability density be negative?

No, the probability density cannot be negative. This is because it represents the likelihood of finding a particle at a specific point, and the probability of finding a particle in a certain location cannot be negative. If the wave function is negative, it will be squared to get the probability density, resulting in a positive value.

How does the normalization of the wave function affect the probability density?

The normalization of the wave function ensures that the total probability of finding a particle in any location is 100%. This means that the probability density at any point will be a valid probability value between 0 and 1. Normalization also allows for the comparison of probability densities between different wave functions.

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