Schrodinger Equations and Probability Density Functions

In summary, the conversation is about a particle described by a normalized wave function and determining its probability of being at a certain distance from the origin. The solution involves using the volume of a spherical shell and finding the maximum value of the probability, which may not necessarily be the same as the maximum value of the function.
  • #1
Epideme
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0

Homework Statement


A Particle is described by the normalized wave function
psi(x,y,z) = Ae^(-alpha[x^2 + y^2 + z^2])
Where A and alpha are real positive constants

a)Determine the probability of finding the particle at a distance between r and r+dr from the origin
hint: use the volume of the spherical shell centered on the origin with inner radius r and thickness dr.
b) For what value of r does the probability in part a) have it's maximum value? Is this the same value of r for which |psi(x,y,z)|^2 is a maximum? explain any differences


Homework Equations


|psi(x)|^2 = 1


The Attempt at a Solution



Got no idea, like all 4 questions I'm posting any help would be appreciated, as i am completely lost
 
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  • #2
the integration measure for sphere is r^2dr d(phi) d(cos theta)

for a) just use the hint, take the probability of finding particle inside sphere with radius r + dr and substract with the probabilitiy to find it inside sphere with radius r.

for b) you know how to find maxima of a given function right?
 

1. What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a partial differential equation that takes into account both the wave-like and particle-like properties of matter.

2. What is the role of the Schrodinger equation in quantum mechanics?

The Schrodinger equation is used to determine the probability of finding a particle in a particular state. It is the basis for understanding the behavior of quantum systems and predicting their future evolution.

3. What is a probability density function?

A probability density function is a mathematical function that describes the probability of a particle being found at a specific location in space. It is used to determine the probability of finding a particle at any given point in space, and it is related to the wave function of a quantum system.

4. How is the probability density function related to the Schrodinger equation?

The probability density function is a solution to the Schrodinger equation. The Schrodinger equation describes how the wave function of a quantum system evolves over time, and the probability density function is derived from the square of the wave function. This means that the probability of finding a particle in a particular state is directly related to the wave function of the system.

5. What are the applications of the Schrodinger equation and probability density functions?

The Schrodinger equation and probability density functions have many applications in quantum mechanics, including predicting the behavior of particles in atomic and molecular systems, understanding the properties of materials, and developing new technologies such as quantum computing. They are also used in other fields such as chemistry, biology, and engineering.

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