A hydrogen molecule (H2) consists of two identical hydrogen atoms separated by a center-to-center distance of approximately 2a = 01 nm and having a total moment of inertia I. (a) Show that the solution to the Schrödinger equation gives wave function Φm=Ae^imϕ where ϕ is the angle of rotation and A

Question

A hydrogen molecule [latex](H_{2} )[/latex] consists of two identical hydrogen atoms separated by a center-to-center distance of approximately [latex]2a = 0.1[/latex] nm and having a total moment of inertia [latex]I[/latex].

(a) Show that the solution to the Schrödinger equation gives wave function [latex]\Phi _{m} =Ae^{im\phi }[/latex] where [latex]\phi[/latex] is the angle of rotation and [latex]A[/latex] is a constant. What values of [latex]m[/latex] are allowed?

(b) At low temperatures the heat capacity of a [latex]H_{2}[/latex] gas is [latex]c_{V} =3k_{B} /2[/latex], where [latex]k_{B}[/latex] is the Boltzmann constant and where each spatial degree of freedom in the [latex]x, y[/latex], and [latex]z[/latex] directions contributes [latex]k_{B} /2[/latex]. At some temperature [latex]T[/latex], thermal energy [latex]k_{B}T[/latex] is high enough to excite rotational modes of the molecule which has the effect of increasing [latex]c_{v}[/latex]. If the rotational energy for total angular momentum state quantum number [latex]l[/latex] is [latex]E=\hbar ^{2} l(l+1)/2I[/latex], estimate the temperature at which this occurs. By how much is [latex]c_{v}[/latex] increased?

Accepted Answer

(a) [latex]\hat{H} \psi_{lm}=\frac{L^{2} }{2I}\psi_{lm}=E_{l} \psi_{lm}=\frac{\hbar ^{2}l(l+1) }{2I} \psi_{lm}[/latex] where [latex]\psi_{lm}=Y^{m}_{l} ( heta ,\phi )=\Phi _{m} (\phi )\Theta^{l}_{m} ( heta )[/latex] and [latex] \frac{\partial}{\partial \phi } \Phi _{m} (\phi )=im\Phi _{m}(\phi )[/latex] so that [latex]\Phi _{m} =Ae^{im\phi }[/latex]. Since we require single-valueness of the function [latex]\Phi _{m} (\phi )[/latex] it follows that the allowed values of [latex]m[/latex] are [latex]m=0, \mp 2,\pm 4,[/latex] ... because the hydrogen atoms are [latex]identical[/latex].

(b) For the ground state angular momentum quantum number [latex]l = 0[/latex]. We seek the characteristic temperature [latex]T=E_{l} /k_{B}[/latex] when it becomes probable that the angular quantum number [latex]l=2[/latex] is excited (remember, from part (a), that [latex]m=\mp 2[/latex] is the first excited state). The moment of inertia of the [latex]H_{2}[/latex] molecule is [latex]I=2ma^{2}[/latex], where the mass [latex]m_{p} =1.67 imes 10^{-27} kg[/latex] can be used for the mass of a hydrogen atom. Putting in the numbers we have

[latex]T=\frac{\hbar ^{2}l(l+1) }{2k_{B}I }=\frac{1.1 imes 10^{-68} imes 2(2+1) }{2 imes 1.4 imes 10^{-23} imes 2 imes 1.7 imes 10^{-27} imes(0.5 imes 10^{-10} )^{2} } =280K[/latex]

and the value of [latex]c_{v}[/latex] is increased by [latex]k_{B}T[/latex] (the two degrees of freedom are the two angles of rotation) to [latex]5k_{B}T/2[/latex].

Question 11.2: A hydrogen molecule (H2) consists of two identical hydrogen ......

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