Quantum chemistry

25 flashcards · Physics · by prod2

1. Time dependent Schrodinger equation?

   iℏδψδt=Hψi\hbar \frac{\delta \psi}{\delta t} = H \psiiℏδtδψ​=Hψ

2. Time independent Schrodinger equation?

   Hψ=EψH \psi = E \psiHψ=Eψ

3. What is the wavefunction ψ\psiψ the function of?

   All degrees of freedom (x, y, z coordinates of every nucleus, electron).

4. Bra-ket notation

   ∫f∗gdτ\int f^{*} g d\tau∫f∗gdτ = "> <f∣g><f|g><f∣g>

5. Hamiltonian operator for momentum

   pqp_{q}pq​ = ℏiδδq\frac{\hbar}{i}\frac{\delta}{\delta q}iℏ​δqδ​

6. Hamiltonian operator for position

   q = q

7. General conditions for valid wavefunctions

   ψ\psiψ has to be: continuous, differentiable not go to infinity anywhere

8. Coulomb's law

   14πϵ0Q1Q2R12\frac{1}{4\pi \epsilon _{0}} \frac{Q_{1}Q_{2}}{R_{12}}4πϵ0​1​R12​Q1​Q2​​

9. Atomic units

   ℏ=1\hbar = 1ℏ=1 me=1m_{e} = 1me​=1 ∣e∣=1|e| = 1∣e∣=1 14πϵ0\frac{1}{4\pi\epsilon _{0}}4πϵ0​1​ =1=1=1

10. Name of the energy unit in atomic units

    Hartree

11. Kinetic energy operator for a 1 dimensional particle + derivation

    E=12mv2E=\frac{1}{2} mv^{2}E=21​mv2 =p22m=\frac{p^{2}}{2m}=2mp2​ pq→p_{q} \rightarrowpq​→ ℏiδδq\frac{\hbar}{i}\frac{\delta}{\delta q}iℏ​δqδ​ Hence the kinetic energy operator H for a 1D particle is −ℏ22mδ2δx2-\frac{\hbar^2}{2m}\frac{\delta^2}{\delta x^2}−2mℏ2​δx2δ2​

12. Kinetic energy operator for a 3 dimensional particle

    H=H=H= −ℏ22m∇2\frac{-\hbar^2}{2m} \nabla^22m−ℏ2​∇2

13. ∇2=?\nabla ^2 = ?∇2=?

    ∇2=\nabla ^2 =∇2= (δ2δx2+δ2δy2+δ2δz2)\left( \frac{\delta ^2}{\delta x ^2} + \frac{\delta ^2}{\delta y ^2} + \frac{\delta ^2}{\delta z ^2} \right)(δx2δ2​+δy2δ2​+δz2δ2​)

14. Properties of operators in this course

    Linear: O^af=aO^f\hat O a f = a \hat O fO^af=aO^f and O^(f+g)=O^f+O^g\hat O (f+g) = \hat O f + \hat O gO^(f+g)=O^f+O^g Hermetian = ^{*}"> <u∣O^∣v>=<v∣O^∣u>∗<u | \hat O | v> = <v | \hat O | u >^{*}<u∣O^∣v>=<v∣O^∣u>∗

15. Properties of hermetian operators

    Their solutions (eigenfunctions) are orthogonal Their eigenvalues are real

16. Probability of finding a particle in a volume

    ∫\int∫ ψ∗ψdτ\psi ^{*} \psi d \tauψ∗ψdτ

17. Normalization of wavefunctions

    = 1"> <ψ∣ψ>=1<\psi | \psi> = 1<ψ∣ψ>=1

18. orthonormal set of functions meaning

    Functions that are: normalized orthogonal

19. Definition of commutative operations

    ABfi=BAfiABf_{i} = BAf_{i}ABfi​=BAfi​

20. Properties of commutative operations

    Common set of eigenfunctions Their eigenvalues can be measured simultaneously

21. Eigenfunctions of an operator form a complete set. All functions depending on the same degrees of freedom/coordinates can be written as a linear combination of the eigenfunctions.

22. If A and B are not commutative, the eigenfunctions of B are not eignefunctions of A.

23. Degenerate functions

    They have the same eignevalue after an operation.

24. Expectation value

    ="> <A>=<A> =<A>= ∫ψ∗A^ψdτ\int \psi ^{*} \hat A \psi d \tau∫ψ∗A^ψdτ (where ψ\psiψ is normalized)

25. Probability of finding a value explanation

    Let A, B be operators that do not commute. Let fif_{i}fi​ be the eigenfunction(s) of A. Let ggg be an eigenfunction of B. Since the eigenfunctions of A form a complete set, ggg can be written as g=g=g= ∑i\displaystyle \sum_{i}i∑​ cific_if_ici​fi​ . If ggg is normalized ( ⟨g∣g⟩=1\lang g | g \rang = 1⟨g∣g⟩=1 ), then ⟨∑icifi∣∑jcjfj⟩\displaystyle\lang \sum_{i} c_if_i | \sum_j c_jf_j \rang⟨i∑​ci​fi​∣j∑​cj​fj​⟩ =∑i∑j\displaystyle =\sum_i\sum_j=i∑​j∑​ cicj⟨fi∣fj⟩c_ic_j \lang f_i | f_j \rangci​cj​⟨fi​∣…