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(Solved): Exercise 4: Consider a particle of mass m that is bouncing vertically and elastically on a reflectin ...
Exercise 4:
Consider a particle of mass m that is bouncing vertically and elastically on a reflecting
hard floor of potential energy V(z) given by
V(z)={(mgz,z>0),(+\infty ,z<=0):}
where g is the gravitational constant.
Consider the ground state trial function \psi _(o)(z,\alpha )=Aze^(-\alpha z)
where \alpha is a parameter and A is the normalization constant.
1- Express A as a function of \alpha .
2- Use the variational method to estimate the ground state energy of this particle.
3- Compare the found expression to the exact one given by
E_(o)^(exact )=2.338((1)/(2)mg^(2)?^(2))^((1)/(3))
Note: \int_0^(+\infty ) x^(n)e^(-\alpha x)dx=(\Gamma (n+1))/(\alpha ^(n+1))=(n!)/(\alpha ^(n+1)) where n is a number and \alpha a positive constant.