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# Proof That the Two-Electron Ground State of a Spin-Independent Hamiltonian Is a Singlet(a) The mean energy of a two-electron system with Hamiltonian (32.3) in the s...

## Question

###### Proof That the Two-Electron Ground State of a Spin-Independent Hamiltonian Is a Singlet(a) The mean energy of a two-electron system with Hamiltonian (32.3) in the state $psi$ can be written (after an integration by parts in the kinetic energy term) in the form:$$E=int d mathbf{r}_{1} d mathbf{r}_{2}left[frac{h^{2}}{2 m}left{left|mathbf{abla}_{1} psiight|^{2}+left|abla_{2} psiight|^{2}ight}+Vleft(mathbf{r}_{1}, mathbf{r}_{2}ight)|psi|^{2}ight]$$Show that the lowest value ( $32.28$ ) assumes over

Proof That the Two-Electron Ground State of a Spin-Independent Hamiltonian Is a Singlet (a) The mean energy of a two-electron system with Hamiltonian (32.3) in the state $psi$ can be written (after an integration by parts in the kinetic energy term) in the form: $$E=int d mathbf{r}_{1} d mathbf{r}_{2}left[frac{h^{2}}{2 m}left{left|mathbf{ abla}_{1} psi ight|^{2}+left| abla_{2} psi ight|^{2} ight}+Vleft(mathbf{r}_{1}, mathbf{r}_{2} ight)|psi|^{2} ight]$$ Show that the lowest value ( $32.28$ ) assumes over all normalized antisymmetric differentiable wave functions $psi$ that vanish at infinity is the triplet ground-state energy $E_{t}$, and that when symmetric functions are used the lowest value is the singlet ground-state energy $E_{mathrm{s}}$ (b) Using (i) the result of (a), (ii) the fact that the triplet ground state $psi_{t}$, can be taken to be real when $V$ is real, and (iii) the fact that $left|psi_{2} ight|$ is symmetric, deduce that $E_{s} leqslant E_{T}$. #### Similar Solved Questions

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31. Function of capsule, slime Jayer 32. Differentiate between slime layer and capsule 33. Define attenuation 34. Structures involved in adherence of Bacteria 35. Explain how dental caries develop 36. Environmental factors that influence skin microflora 37. LDso 38. Phagocytosis 39. Septicemia 40 . ...
##### 1 range Appoodimatoty 333 the Io current 3 01 What equivalent M 341 Hi hesiseuce that the between E 1 points LL ray incident. and ~fall makes for the case 1 the the range with figure when a [ below (R with the normal if the second medium given count is repeated 0f 25,0 N degrees What should fall in 42 in a medium the total times? the
1 range Appoodimatoty 333 the Io current 3 01 What equivalent M 341 Hi hesiseuce that the between E 1 points LL ray incident. and ~fall makes for the case 1 the the range with figure when a [ below (R with the normal if the second medium given count is repeated 0f 25,0 N degrees What should fall in ...
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The behavior wc observed in the previous question is perhaps best understood by looking at different-sized squares and comparing the added area WC get as we move from one size tO another: The added area is 3 , then 5, then square inches as we add inch to the side length 1x 1 2 x 2 3 x 3 4x 4 The blu...
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