CHM345_HW7_13

Now verify that whe heisenberg uedition the ground

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Unformatted text preview: HermiteH Ν, 2 DD Out[290]= ΩΜ 12 x Exp 12 x ^2 2 2 HermiteH Ν, x , x , x, Out[289]= ΩΜ , ΩΜ 12 x Exp , Assumptions ΩΜ ΩΜ 12 x ^2 0 && Ω Μ 2 , 0 0 ΜΩ 4 In[293]:= Out[293]= ΣPx ,1 1 2 ΜΩ 4 ΜΩ Printed by t olfram Mathematica Student ncertainty principle is satisfied for Exercise 4. Now verify that Whe Heisenberg UEdition the ground vibrational state (Υ = 0) and the first excited vibrational state (Υ = 1). Recall, 9 10 CHM345_HW7_13.nb Exercise 4. Now verify that the Heisenberg Uncertainty principle is satisfied for the ground vibrational state (Υ = 0) and the first excited vibrational state (Υ = 1). Recall, Σx Σpx (4) 2 For Ν=0 ΜΩ 2ΜΩ 2 Out[300]= 1 2 2 ΜΩ ΜΩ For Ν=1 In[302]:= 1 2 Out[302]= 1 ΜΩ 1 4 ΜΩ 2 ΜΩ ΜΩ Exercise 5. Equipartition Theorem. Using some of the same integrals from the previous problems prove, for the Υ = 0 and Υ = 1 vibrational states, the general Equipartition Theorem T Υ V EΥ Υ 2 Printed by Wolfram Mathematica Student Edition (5) CHM345_HW7_13.nb V Υ x2 12k 11 Υ Ν12 ΜΩ Ω Ν12 12k 12 1 2 EΝ TΥ "half potential" & "half kinetic" Exercise 6. The Hermite Polynomials display a number of general relations among themselves. One of the more useful relations is: HΥ 1 2 Ξ HΥ Ξ Ξ 2 Υ HΥ 1 Ξ (6) Rearranging, we have 1 Ξ HΥ Ξ 2 HΥ 1 Υ HΥ Ξ 1 Ξ (7) Exploiting this relation and the orthonormalilty of the vibrational eigenstates, show that a. for all Υ x Υ 0 (8) b. for all Υ x2 Υ 1 Υ ΩΜ (9) 2 c. Now prove that for all Υ Vx Ω...
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