Quantum Mechanics and Atomic Physics – Flashcards
Unlock all answers in this set
Unlock answersquestion
probability of position
answer
|Γ|²
question
Expectation value of A
answer
question
Hermitian operator
answer
∫ƒ(x)*(Ag(x))dx = ∫(Af(x))g(x)dx
question
Facts about hermitians (2)
answer
All their eigenvalues are real Eigenfunctions corrisponding to different eigenvalues are orthogonal
question
measurement of an operator
answer
always returns an eigenvalue, and if discrete the measurement of A is quantized
question
sqrt of probability of a measurement
answer
Cn = ∫fn(x)*Γ(x,t)dx where fn is the orthonormal basis
question
expectation value if the probabilities are known
answer
question
inner product of |a> and |b>
answer
question
dirac notation for the hermitian conjugate operator
answer
question
hermitian conjugate
answer
the conjugate operator that would provide the same answer when the operator is applied to the other vector in the inner product
question
hermitian operator
answer
hermitian operators are equal to their hermitian conjugate
question
Hilbert Space
answer
Infinite dimensional vector space
question
conjugate vectors in Hilbert Space
answer
<b| = (transposeb)*
question
conjugate operators in Hilbert Sfpace
answer
A⁺ = (transposeA)*
question
commutator
answer
[a,b] = ab-ba
question
commutator of x and p
answer
{x,p} = ihbar
question
commutator identity for [AB,C]
answer
[AB,C] = A[B,C] + [A,C]B
question
commutator identity for [A,BC]
answer
[A,BC] = [A,B]C + B[A,C]
question
Uncertainty principle
answer
σa²σb²≥((1/2i))²
question
Heisenburg uncertainty principle
answer
σxσp≥hbar/2 and is usually in the range of ∆x∆p≈hbar
question
nodes in quantum mechanics
answer
correspond to the energy level starting at 0 for n=0
question
One dimension harmonic osc hamiltonian
answer
H = hbarw(a⁺a+1/2)
question
harmonic osc [a,a+]
answer
[a,a⁺]=1
question
harmonic osc energy levels
answer
En=hbarw(n+.5)|n> from 0→∞
question
lowering operator a
answer
a|n> = √(n)|n-1>
question
raising operator a+
answer
a⁺|n> = √(n+1)|n+1>
question
viral theorem inc. harmonics
answer
==En/2
question
three dimensional harmonic Energies
answer
En=(n₁+n₂+n₃+1.5)hbarw
question
infinite square well wavefnctn
answer
γ=sqrt(2/a)sin(nπx/a)
question
infinite square well energies
answer
n²π²hbar²/2ma²
question
free particle wavefnctn
answer
γ=e^(±ikx)
question
energy of free particle
answer
E=hbar²k²/2m where k=p/hbar
question
scattering wavefnctns
answer
Before impact with V: Ae^(ikx) + Be^(-ikx) During impact with V: something After impact with V: Ce^(ik'x)
question
probabilities of reflection and transmition
answer
R=|B|²/|A|²; T=|C|²/|A|²; R+T=1
question
normalization in 3D
answer
radial and angular pieces are normalized seperately
question
ang momentum operators
answer
Lx=yPz-zPy; Ly=zPx-xPz; Lz = xPy-yPx
question
total angular momentum operator L^2
answer
L²=-hbar²del²
question
z angular momentum operator
answer
Lz=-ihbard/d∅
question
L^2 acting on angular wavefunction
answer
L²Y=l(l+1)hbar²Y
question
Lz acting on angular wavefunction
answer
LzY=mhbarY
question
allowed values of m given l
answer
m=l,l-1,...-l
question
phi dependence of Y
answer
e^(im∅)
question
Hamiltonian for hydrogen
answer
H = -hbar²del²/2µ + e²/(4πε₀r)
question
Bohr radius
answer
a=4πε₀hbar²/µe^2
question
wavefunction for hydrogen atom approx
answer
psy1 α e^(-r/a)
question
energy for a hydrogenic atom
answer
hbar²/2µa² = E1
question
energy for hydrogen ground state
answer
13.6 eV
question
energy for following hydrogenic states
answer
E1/n²
question
frequency of emitted light from hydrogenic atoms
answer
f α 1/n₂² - 1/n₁²
question
fine structure constant
answer
1/137
question
pauli spin matrix for x
answer
[0,1;1,0]
question
pauli spin matrix for y
answer
[0,-i;i,0]
question
pauli spin matrix for z
answer
[1,0;0,-1]
question
addition of s
answer
s values dont add, but possible values range from s+s' to |s-s'|
question
addition of ms
answer
mstot = ms +ms'
question
Total spin operator for addition
answer
S = S₁ + S₂, can either be - or +
question
spin lowering operator on two electrons in the s=1,m=1 state
answer
S⁻|↑>|↑> = (S⁻₁|↑>|↑> + S⁻₂|↑>|↑> = hbar(|↓>|↑>+|↑>|↓>), now in s=1,m=0 state
question
singlet state of two electrons
answer
s=0,m=0; |↑>|↓>-|↓>|↑>
question
wavefunctions of bosons
answer
symmetric (constructive)
question
wavefunctions of fermions
answer
antisymmetric (destructive)
question
Pauli Exclustion principle
answer
No two identical fermions can occupy the same state
question
first order Hamiltonian perturbation
answer
H = H₀+λH';
question
first order energy perturbation
answer
En = En₀ + λ
question
2nd order energy perturbation
answer
E₀+λ²∑||²/En-Em
question
simplification of
answer
H'mn, in other words, the off diagonal elements of H'mn
question
degenerate actions
answer
Diagonalize the peturbation in the subspace of degenerate states
question
variational principle
answer
cannot be less than the unperturbed ground state
question
Adiabatic theorem of QM
answer
If starting in H, the particle will end up in the /corresponding/ eigenstate of H'
question
Bohr model
answer
no radiation in specific rings around atom, L=nhbar
question
Fine structure
answer
relativistic correction for the electron + spin orbit coupling, intro to J=L+S, order α² smaller
question
Lamb shift
answer
accounts for J degeneracy; order α³ smaller
question
Spin-Spin coupling
answer
magnetic dipole dipole interaction between protons and electrons; order α² smaller;
question
orbital number per energy shell
answer
2n²
question
states per orbital
answer
2(2l+1)
question
Stark effect
answer
E field on Hydro, only m=0 is split, ∆H = eE.r
question
Zeeman effect
answer
B field on Hydro, ∆H=(e/2m)(L+2S).B; e/2m is known as the gyromagnetic ratio
question
Selection rules
answer
∆m=±1 or 0; ∆l=±1; except from 2s→1s
question
Blackbody radiation intensity
answer
I α (hw³/c²)(1/(e^(hbarw/kbT) -1)
question
Power per area pf Blackbody
answer
dP/dA α T⁴
question
Peak of BB spectrum (Wein's displacement Law)
answer
λmax = (2.9x10⁻³ K.m)/T
