Schrodinger's Equation Flashcards
What is Schrodinger’s equation? (TDSE)
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).
When are the stationary states for the wavefunction Ψ(x,y,z,t) in the Schrodinger equation?
Ψ(x,y,z,t) = ф(x,y,z,t)exp(-iEn*t/ћ)
What can we write V(x,y,z) as if it is Spherically symmetric?
V(x,y,z) = V(r)
What can we write about the wavefunction Ψ(r, θ, Ф) for the TISE?
H(hat)ф(nlm)(r, θ, Ф) = Enф(nlm)(r, θ, Ф), and ф(nlm)(r, θ, Ф) = R(r)*ψ(lm)(θ, Ф)
Which 3 values commute in the TISE and what can we say about these?
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
What is L(hat)^2 times the wavefunction ψ(θ, Ф) equal to?
L(hat)^2 ψ(lm)(θ, Ф)= l(l+1)*ћ^2 *ψ(lm)(θ, Ф)
What is Lz(hat) timesthe wavefunction ψ(θ, Ф) equal to?
Lz(hat)ψ(lm)(θ, Ф) = mћ*ψ(lm)(θ, Ф)
What is ф(r, θ, Ф, σ) equal to?
ф(nlm,m(s))(r, θ, Ф, σ) = ф(r, θ, Ф)*X(1/2,m(s))(σ)
What does X(1/2,m(s))(σ) equal for electrons?
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
What is the equation for En, the energy of a state?
En = -13.6eV/n^2
What is the equation for the potential V(r) of an atom?
V(r) = -e^2/4πε0r
What does the Pauli exclusion principle state?
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))
What is the potential like in an atom?
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.
What does this non-coulombic potential do?
Removes some degeneracy of the energy levels with respect to the orbital angular momentum, l.
How can we work out the order of the energies?
Can be predicted from considering the average probability of an electron in an orbital being found close to the nucleus.
How do we work out the order of electrons in an atom?
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.
What is the total kinetic energy of an atom equal to?
KE of the electrons + KE of the nuclei + interaction between electrons and nuclei + interaction between nuclei + interaction between electrons + interaction with external fields
How can we write the KE of the electrons in terms of the Schrodinger equation with n electrons, N nuclei?
sum from i=1 to n of (-ћ^2/2m *∇i^2)
How can we write the KE of the nuclei in terms of the Schrodinger equation with n electrons, N nuclei?
sum from I=1 to N of (-ћ^2/2m(I) *∇(I)^2)
How can we write the KE of the electrons interacting with the nuclei in terms of the Schrodinger equation with n electrons, N nuclei?
-sum from i=1 to n * sum from J = 1 to N of (z(J)*e^2)/(4πε0(|r(i)-R(J)|)
How can we write the KE of the electrons interacting with the electrons in terms of the Schrodinger equation with n electrons, N nuclei?
sum from i = 1 to n * sum from j = 1 to n (i=/j) of e^2/(4πε0*|r(i)-r(j)|)
How can we write the KE of the nuclei interacting with the nuclei in terms of the Schrodinger equation with n electrons, N nuclei?
sum from J=1 to N * sum from J=/ i to N of (z(I)Z(J)e^2)/(4πε0*|R(I)-R(J)|)
How can we write the KE of the interaction with external field in terms of the Schrodinger equation with n electrons, N nuclei?
V(ext)(…)
What is the first approximation we make to look at larger systems?
Neglect KE of nuclei - freeze the nuclear positions, and solve electronic problem. Use this to get the potential for nuclei.
What is the example of a covalent bond?
H2 molecule: model with 1 electron in ground state centred on proton 1 and the other centred on proton 2.
For the H2 example, what do we need to do?
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/
What does the graph of ground state energy against R(12) look like, where R12 is the distance between the nuclei?
Like the strong nuclear force graph, where the turning point is the bond length R(b)
How can we use the electrons shell letter thingys to show an ionic bond for e.g NaCl?
3s1 from Na goes to 3p5 on Cl so it gives an electron so they both have filled outer shells.
