Evaluate the probability that a particle trapped in a box L wide can be found between 0.45L and 0.55L for the ground and first excited states.
Question
Evaluate the probability that a particle trapped in a box L wide can be found between 0.45L and 0.55L for the ground and first excited states.
Solution
The probability of finding a particle in a certain region in a box is given by the square of the wave function integrated over that region.
For a particle in a box, the wave functions for the ground state (n=1) and first excited state (n=2) are given by:
ψ_n(x) = sqrt(2/L) sin(nπx/L)
- Ground State (n=1):
The probability of finding the particle between 0.45L and 0.55L is given by the integral from 0.45L to 0.55L of |ψ_1(x)|^2 dx.
|ψ_1(x)|^2 = 2/L * sin^2(πx/L)
Integrating this from 0.45L to 0.55L gives the probability for the ground state.
- First Excited State (n=2):
Similarly, the probability for the first excited state is given by the integral from 0.45L to 0.55L of |ψ_2(x)|^2 dx.
|ψ_2(x)|^2 = 2/L * sin^2(2πx/L)
Integrating this from 0.45L to 0.55L gives the probability for the first excited state.
These integrals can be solved using standard techniques of integration. The exact values will depend on the specific values of L, but the procedure outlined above gives the general method for solving this type of problem.
Similar Questions
A particle is confined in an one dimensional box (0<x<L). The probability of finding the particle in the interval 0<x<L isa.1b.depends on the quantum number nc.depends on the energy of the particle
What is the probability of finding the particle in the region [0, L/2]. You may calculate the answer directly, or write itdown and explain why it must be true (but you need a mathematical argument !), don’t just write the answer
A particle is confined in a 1D box (0<x<L). The ratio of the energies of the particle in state n=3 to state n=2 isa.9/4b.1/4c.4
ompute the energy difference between the ground state and first excited state for an electron in a one-dimensional rigid box of length 10–8 cm. Given m = 9.1 × 10–31 kg and h = 6.626 × 10–34 J sec
Boundary conditions and normalization determine the wave functions. Considering a particle trapped in a box with infinitely hard walls, comment on the above sentence and find wave function nthat corresponds to various energy levels. Where is this particle most likely to be in the lowest energy level and what will be the probability of finding the particle at this location in the next higher energy state?
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.