Schrodinger's Equation

Question 1

What is Schrodinger’s equation? (TDSE)

Answer 1

iћ d/dt(Ψ(x,y,z,t)) = (-iћ^2/2m ∇^2 + V(x,y,x)Ψ(x,y,z,t), where the part in brackets on the right ride is the hamiltonian H(hat).

Question 2

When are the stationary states for the wavefunction Ψ(x,y,z,t) in the Schrodinger equation?

Answer 2

Ψ(x,y,z,t) = ф(x,y,z,t)exp(-iEn*t/ћ)

Question 3

What can we write V(x,y,z) as if it is Spherically symmetric?

Answer 3

V(x,y,z) = V(r)

Question 4

What can we write about the wavefunction Ψ(r, θ, Ф) for the TISE?

Answer 4

H(hat)ф(nlm)(r, θ, Ф) = Enф(nlm)(r, θ, Ф), and ф(nlm)(r, θ, Ф) = R(r)*ψ(lm)(θ, Ф)

Question 5

Which 3 values commute in the TISE and what can we say about these?

Answer 5

H(hat), Lz(hat) and L(hat)^2 commute, so [H(hat), L(hat)^2] = [H(hat), Lz(hat)] = [L(hat)^2, Lz(hat)] = 0

Question 6

What is L(hat)^2 times the wavefunction ψ(θ, Ф) equal to?

Answer 6

L(hat)^2 ψ(lm)(θ, Ф)= l(l+1)*ћ^2 *ψ(lm)(θ, Ф)

Question 7

What is Lz(hat) timesthe wavefunction ψ(θ, Ф) equal to?

Answer 7

Lz(hat)ψ(lm)(θ, Ф) = mћ*ψ(lm)(θ, Ф)

Question 8

What is ф(r, θ, Ф, σ) equal to?

Answer 8

ф(nlm,m(s))(r, θ, Ф, σ) = ф(r, θ, Ф)*X(1/2,m(s))(σ)

Question 9

What does X(1/2,m(s))(σ) equal for electrons?

Answer 9

X(1/2,m(s))(σ) = matrix (1,0) for m(s)=1/2 or X(1/2,m(s))(σ) = matrix(0,1) for m(s) = -1/2

Question 10

What is the equation for En, the energy of a state?

Answer 10

En = -13.6eV/n^2

Question 11

What is the equation for the potential V(r) of an atom?

Answer 11

V(r) = -e^2/4πε0r

Question 12

What does the Pauli exclusion principle state?

Answer 12

Many electron state must be antisymmetric (e.g. 2 electron state: ф(r1, θ1, Ф1, σ1;r2, θ2, Ф2, σ2) = -ф(r2, θ2, Ф2, σ2;r1, θ1, Ф1, σ1))

Question 13

What is the potential like in an atom?

Answer 13

Different from the straight coulomb potential because there is a potential set up by the positively moving charged nucleus and symmetric cloud of the negative charge from the other Z-1 electrons.

Question 14

What does this non-coulombic potential do?

Answer 14

Removes some degeneracy of the energy levels with respect to the orbital angular momentum, l.

Question 15

How can we work out the order of the energies?

Answer 15

Can be predicted from considering the average probability of an electron in an orbital being found close to the nucleus.

Question 16

How do we work out the order of electrons in an atom?

Answer 16

Look at atomic number z, and with the s, p, d states etc have to the power so 1s^2, 2s^2, 2p^6, etc, and the powers must add up to the atomic number Z.

Question 17

What is the total kinetic energy of an atom equal to?

Answer 17

KE of the electrons + KE of the nuclei + interaction between electrons and nuclei + interaction between nuclei + interaction between electrons + interaction with external fields

Question 18

How can we write the KE of the electrons in terms of the Schrodinger equation with n electrons, N nuclei?

Answer 18

sum from i=1 to n of (-ћ^2/2m *∇i^2)

Question 19

How can we write the KE of the nuclei in terms of the Schrodinger equation with n electrons, N nuclei?

Answer 19

sum from I=1 to N of (-ћ^2/2m(I) *∇(I)^2)

Question 20

How can we write the KE of the electrons interacting with the nuclei in terms of the Schrodinger equation with n electrons, N nuclei?

Answer 20

-sum from i=1 to n * sum from J = 1 to N of (z(J)*e^2)/(4πε0(|r(i)-R(J)|)

Question 21

How can we write the KE of the electrons interacting with the electrons in terms of the Schrodinger equation with n electrons, N nuclei?

Answer 21

sum from i = 1 to n * sum from j = 1 to n (i=/j) of e^2/(4πε0*|r(i)-r(j)|)

Question 22

How can we write the KE of the nuclei interacting with the nuclei in terms of the Schrodinger equation with n electrons, N nuclei?

Answer 22

sum from J=1 to N * sum from J=/ i to N of (z(I)Z(J)e^2)/(4πε0*|R(I)-R(J)|)

Question 23

How can we write the KE of the interaction with external field in terms of the Schrodinger equation with n electrons, N nuclei?

Answer 23

V(ext)(…)

Question 24

What is the first approximation we make to look at larger systems?

Answer 24

Neglect KE of nuclei - freeze the nuclear positions, and solve electronic problem. Use this to get the potential for nuclei.

Question 25

What is the example of a covalent bond?

Answer 25

H2 molecule: model with 1 electron in ground state centred on proton 1 and the other centred on proton 2.

Question 26

For the H2 example, what do we need to do?

Answer 26

Construct 2 electron wavefunction which is antisymmetric: Ψ(r1,r2) = ф(100)(r1)*ф(100)(r2) -> leads to waveform joining and making a waveform which looks like a camels humps/

Question 27

What does the graph of ground state energy against R(12) look like, where R12 is the distance between the nuclei?

Answer 27

Like the strong nuclear force graph, where the turning point is the bond length R(b)

Question 28

How can we use the electrons shell letter thingys to show an ionic bond for e.g NaCl?

Answer 28

3s1 from Na goes to 3p5 on Cl so it gives an electron so they both have filled outer shells.