Particle in a Box: Calculate Probability at Centre

In summary, the conversation discusses the calculation of the probability of finding a particle in a one-dimensional box with infinite height and width of 10 Angstroms. The particle is in its ground state, with a wave function of \psi _{1}=\sqrt{\frac{2}{L}}sin \frac{\pi x}{L}. The probability amplitude is found by squaring the wave function, and the probability between two specific points is calculated by taking the integral of the squared wave function between those points. The conversation then focuses on finding the probability within an interval of 1 Angstrom at the centre of the box, with the final conclusion being that the limits for integration should be between 4.5 and 5.5 Ang
  • #1
roshan2004
140
0

Homework Statement



A particle is moving in a one dimensional box of infinite height of width 10 Angstroms. Calculate the probability of finding the particle within an interval of 1 Angstrom at the centre of the box, when it is in its state of least energy.

Homework Equations



[tex]
\psi _{n}=\sqrt{\frac{2}{L}}sin \frac{n\pi x}{L}
[/tex]

The Attempt at a Solution


The wave function of the particle in the ground state (n=1) is [tex]\psi _{1}=\sqrt{\frac{2}{L}}sin \frac{\pi x}{L}[/tex]. Now, what should I do ?
 
Physics news on Phys.org
  • #2
How does one find the probability amplitude in QM?
 
  • #3
Square of its wavefunction. I got <tex> \frac{2}{L}sin^2 \frac{\pi x}{L}</tex> Now...
 
  • #4
Okay, now how do you find the probability on the interval [4 Angstroms,6 Angstroms]?
 
  • #5
Why between 4 and 6 angstroms ?
 
  • #6
[STRIKE]Why do you think it's between 4 and 6 angstroms?
[/STRIKE]
EDIT: Err rather I believe it should be from 4.5 to 5.5 angstroms...
 
  • #7
probability of finding the particle between x & x+dx is [tex]{|\Psi|}^{2}[/tex]

probability of finding the particle between x=a and a=b is [tex]\int_{a}^{b}{|\Psi|}^{2}dx[/tex]
 
  • #8
What are the limits I should use for the integration?
 
  • #9
Find a and b for "an interval of 1 Angstrom at the centre of the box".
 

FAQ: Particle in a Box: Calculate Probability at Centre

What is a particle in a box?

A particle in a box is a simplified model used in quantum mechanics to study the behavior of a particle confined to a one-dimensional space. It is often used to explain the behavior of electrons in atoms or molecules.

What is the probability at the centre for a particle in a box?

The probability at the centre for a particle in a box is the likelihood of finding the particle at the exact centre of the box. It is determined by solving the Schrödinger equation for the particle in the box, which gives a probability density function that is highest at the centre of the box.

How do you calculate the probability at the centre for a particle in a box?

To calculate the probability at the centre for a particle in a box, you need to first determine the wavefunction for the particle in the box using the Schrödinger equation. Then, you can square the wavefunction to get the probability density function. Finally, you can plug in the value for the centre of the box into the probability density function to get the probability at the centre.

What factors affect the probability at the centre for a particle in a box?

The probability at the centre for a particle in a box is affected by the size of the box, the potential energy of the box, and the mass of the particle. A larger box or a higher potential energy will result in a lower probability at the centre, while a smaller box or a lower potential energy will result in a higher probability at the centre. The mass of the particle also affects the probability at the centre, with heavier particles having a lower probability at the centre compared to lighter particles.

Why is calculating the probability at the centre important in quantum mechanics?

Calculating the probability at the centre for a particle in a box is important in quantum mechanics because it allows us to understand the behavior of particles in confined spaces. This model can also be used to study more complex systems, such as atoms and molecules, and can provide insight into their properties and behaviors. Additionally, the probability at the centre is a key factor in determining the energy levels of a particle in a box, which has significant implications in many areas of physics and chemistry.

Similar threads

Particle in a box that suddenly expands
Replies
15
Views
3K
Probability of finding a particle in the right half of a rectangular potential well
Replies
16
Views
2K
Quantum Well Centred at the Origin Doubled in Size at time t=0. Symmetry seems to render the question incorrect
Replies
4
Views
1K
Most Probable energy after infinite square well expands
Replies
2
Views
783
Spatial Wave function of two indistinguishable particles
Replies
26
Views
3K
Proton in a 1D Box: Energy, Probability, Speed
Replies
24
Views
2K
Zero-point energy of diatomic hydrogen (particle in a box)
Replies
3
Views
1K
Proving the Commutation Relation for Quantized Boson in a One-Dimensional Box
Replies
1
Views
862
Entropy of disks in a 2d box
Replies
6
Views
1K
Particle in a box: Finding <T> of an electron given a wave function
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top