Wave mechanics: the ground state and excited state of nitrogen attom

[noblegas]

Homework Statement

a) To get a wave function for a situation in which the energy is close to $$E_0$$ and the atom is almost certainly in one of the minama of the potential energy , consider the functions

$$\varphi_t(x)=[(\varphi_0(x)+\varphi_1(x))/(2^(1/2)))$$,
$$\varphi_b(x)=[(\varphi_0(x)-\varphi_1(x))/(2^(1/2)))$$

a) Using the fact that $$E_0$$ and $$E_1$$ are very nearly equal , show that , with appropriate choices of phases for the round state and first excited state wave functions $$\varphi_t(x)$$ and $$\varphi_1$$, the function $$\varphi_t$$ is apprecialbly different from zero on only one side of the triangle, and $$\varphi_b(x)$$ is appreciabley different from zero only on one side of the triangle, and $$\varphi_b(x)$$ is appreciably different from zero only the other side

b) show that $$\varphi_t$$ and$$\varphi_b$$ are probably normalized if $$\varphi_0$$ and $$\varphi_1$$ are normalized

c) If the wave function for the molecule is $$\varphi_t$$ what is the probablity that the result of the measurement of energy will be the ground state value $$E_0$$

d) Given the initial condition that the wave function is $$\varphi_t$$ at time t=0 , find the wave function as a function of the time in terms of eignefunctions and eigenvalues

Homework Equations

$$\varphi_t(x)=[(\varphi_0(x)+\varphi_1(x))/2^(1/2))$$,
$$\varphi_b(x)=[(\varphi_0(x)-\varphi_1(x))/2^(1/2))$$, possibly

[tex]\varphi(t)=\varphi_\pi,_n*exp(-iE_nt/$$\hbar$$) and$$dP/dx=|\varphi^2|$$

The Attempt at a Solution

a) should I take the commutator of $$[\varphi_b(x),\varphi_t(x)]$$ which equals zero.

b) I probably need to take the conjugate of $$\varphi_t$$ and $$\varphi_b$$ ; Not sure how to get $$\varphi_0$$ and $$\varphi_1$$

c) , Find the probability density of $$\varphi_t$$ ?

d)
$$\varphi_t(x)=[(\varphi_0(x)+\varphi_1(x))/2^(1/2))$$,=> $$\varphi_t(x)=[(\varphi_0(x)*exp(-E_0t*i/\hbar) +\varphi_1(x))*exp(-E_1*t*i/\hbar)/2^(1/2))$$?

 

[Jhero]

Hello!

First of all, great job on starting to work through the problem. I'll try to help you out with your attempted solutions:

a) Taking the commutator of [\varphi_b(x),\varphi_t(x)] is not necessary for this problem. Instead, you can use the fact that E_0 and E_1 are very close to each other to simplify the expressions for \varphi_t(x) and \varphi_b(x). You can then use trigonometric identities to show that \varphi_t(x) and \varphi_b(x) are only appreciably different from zero on one side of the triangle.

b) To show that \varphi_t and \varphi_b are probably normalized, you can use the fact that \varphi_0 and \varphi_1 are normalized and the expressions for \varphi_t(x) and \varphi_b(x) that you obtained in part a.

c) To find the probability that the result of the measurement of energy will be the ground state value E_0, you can use the probability density function, which is given by |\varphi_t(x)|^2. You can then integrate this over the region where \varphi_t is appreciably different from zero to find the probability.

d) To find the wave function as a function of time, you can use the time evolution operator, which is given by exp(-iHt/\hbar), where H is the Hamiltonian operator. You can then express the initial wave function \varphi_t at t=0 in terms of the eigenfunctions and eigenvalues of the Hamiltonian (which are \varphi_0 and \varphi_1, respectively) and use the time evolution operator to find the wave function at any time t.

Hope this helps! Keep up the good work.