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

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Homework Statement



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

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

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

b) show that [tex]\varphi_t[/tex] and[tex] \varphi_b [/tex] are probably normalized if [tex]\varphi_0[/tex] and [tex]\varphi_1[/tex] are normalized

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

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



Homework Equations





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

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

The Attempt at a Solution



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

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

c) , Find the probability density of [tex]\varphi_t[/tex] ?

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