# Particle in a Box

A particle is constrained to lie along a one-dimensional line segment between 0 and a. The probability of finding the particle along this line segment is seen in attached file.

A. Find the proportionality constant, ie., normalize the function.

B. Calculate the expectation values of X and X^2 for n=1 and n=2.

C. Find the Variance and standard deviation for a particle with n=1 and 2.

D. What is the probability of finding the particle between 0 and a/2 when n=1 and 2.

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#### Solution Preview

Please find the attached solution.

A particle is constrained to lie along a one-dimensional line segment between 0 and a. The probability of finding the particle along this line segment is :

A. Find the proportionality constant, i.e., normalize the function.

B. Calculate the expectation values of X and X^2 for n=1 and n=2.

C. Find the Variance and standard deviation for a particle with n=1 ...

#### Solution Summary

This in-depth solution contains step-by-step calculations with explanations to determine the proportionality constant, expectation values, variance, standard deviation and probability. All formulas and workings are shown in structured manner.

Particle in a Box for State Particle

Consider a particle of mass m in a box of length a. Assume the state of the particle at time t=0 is:

Where are the particle in a box stationary states with energy

a. What is the probability a measurement of the energy will result in

b. What is the expectation (average) value for the energy (<E>) in the state ?

c. What is the probability a single measurement will yield the value E=<E>?

d. Write down an expression for the time-dependent wavefunction for this state, , in terms of the mass of the particle and the length of the box?