130_notes.dvi

Thus we have a large effective spin-spin interaction entirely due to electron repulsion. There is a large difference in energy b ...

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25.4 The Variational Principle (Rayleigh-Ritz Approximation) Because the ground state has the lowest possible energy, we can var ...

We wish to show thatE′errors are second order inδψ ⇒ ∂E ∂ψ = 0 at eigenenergies. To do this, we will add a variable amount of an ...

Minimize the energy. 〈ψ|H|ψ〉= ∫ d^3 r 1 d^3 r 2 φ∗ 100 (~r 1 )φ∗ 100 (~r 2 ) [ p^21 2 m − Ze^2 r 1 + p^22 2 m − Ze^2 r 2 + e^2 | ...

25.6 Examples 25.6.1 1D Harmonic Oscillator Use ψ= ( a^2 −x^2 ) 2 |x|≤a andψ= 0 otherwise as a trial wave function. Recall the a ...

= [ −a^2 ̄h^2 2 m + 1 2 mω^2 ] ∞∫ −∞ x^2 e−ax 2 dx ∞∫ −∞ e−ax^2 dx + [ ̄h^2 a 2 m ] ∫∞ −∞ e−ax 2 dx= √ π a = √ πa−^1 /^2 − ∫∞ −∞ ...

1 |~r 1 −~r 2 | = 1 √ r 12 +r^22 − 2 r 1 r 2 cosθ Do thedΩ 1 integral and prepare the other. ∆Egs= 4 π π^2 e^2 ( Z a 0 ) 6 ∫∞ 0 ...

= − Ze^2 a 0 [ 3 2 2 2 1 2 1 2 + 2 2 2 1 2 1 2 + 2 1 2 1 2 − 2 1 1 1 1 ] ∆Egs = − Ze^2 a 0 [ 3 8 + 4 8 + 4 8 − 16 8 ] = + 5 8 Ze ...

A Helium atom has two electrons bound to aZ= 2 nucleus. We have to add the coulomb repulsion term (between the two electrons) t ...

26 Atomic Physics This material is covered inGasiorowicz Chapter 19,and inCohen-Tannoudji et al. Com- plementAXIV. 26.1 Atomic S ...

The effect of screening not only breaks the degeneracy between states with the samenbut different ℓ, it even moves the 6s state, ...

states in which the totalsand the totalℓare good quantum numbers are best for minimizing the overlap of electrons, and hence the ...

Z El. Electron Configuration^2 s+1Lj Ioniz. Pot. 1 H (1s)^2 S 1 / 2 13.6 2 He (1s)^21 S 0 24.6 3 Li He (2s)^2 S 1 / 2 5.4 4 Be H ...

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The perodic table of elements is based on the fact that atoms with the same number of electrons outside a closed shell have simi ...

Even with the strong nuclear force, a shell model describes important features of nuclei. Nuclei have tightly bound closed shell ...

Another feature of nuclei not shown in the table is that thespin-spin force very much favors nucleons which are paired. So nucle ...

mℓ e 1 ↑ 0 -1 s= ∑ ms=^12 ℓ= ∑ mℓ= 1 26.6.2 Carbon Ground State Carbon, withZ= 6 has the 1S and 2S levels filled givingj= 0 as a ...

The ground state hass= 1 andℓ= 1 as we predicted. Other states labeled 2pare the ones that Hund’s first two rules determined to ...