PCHEM Exam 1- Units 2,3,4, & 5
Hermite polynomials
(Hn (ɑ^1/2x). These are polynomial functions where Hn is a nth degree polynomial in Xi
linear operators
(ex. - operator A^) a special property of these operators is that a linear combination of two eigenfunctions of the operator with the same eigenvalue is also an eigenfunction of the operator
quantum number
(symbolized by the letter n) describes the value of an energy level. For particle in a box, n cannot equal 0, it starts at n=1. They appear natural in the Schrödinger equation, not ad hoc like in Plank model or Bohr model
normalization constant (B)
= (2/a)^1/2. Find B by setting the condition that the function Ψ(x) is normalized
radial node
= n-l-1 // point at which probability is zero of find electrons at a radius, r and probability and wave function are at zero on plotted graph // r=0 is not a node // when R(r) = 0
probability density*****
A function that describes the relative likelihood for a sample to be in a certain region. The probability of finding a particle is always positive. Generally, probability density increases as n increases.
observable
A measurable dynamical variable. Usually the eigenvalue(Energy) of an eigenfunction.
commutation
A property of operators in which A^B^f(x) is compared to B^A^f(x). (you use A^ as the operator for B^f(x) and then use B^ as the operator for A^f(x). Operators do not usually commute - they do not normally equal the same thing)
wave functions
AKA state function. has the important property that Ψ*(x)Ψ(x)dx is the probability that the particle lies in the interval dx, located at the position x.
moment of inertia****
I = μr^2. the rotational analog of mass for linear motion
average value
If a system is in a state described by a normalized wave function Ψ, then the average value, <a>, of the observable corresponding to A^ (operator) is given by : ∫ Ψ*A^(operator)Ψ dx
separable Hamiltonian
If the Hamiltonian operator can be written as a sum of terms involving different coordinates(Ex. x,y,z), then the eigenfunctions of H ^ is a product is a product of eigenfunctions of each operator, and the eigenvalues of H ^ is a sum of eigenvalues of each operator constituting the sum.
orthonormal
If the two wave functions are normalized (the integral of the product of the wave function and the complex conjugate = 1) then the 2 functions are orthonormal
Eigenvalue-eigenfunction relationship
In the relationship, there is an operator for the eigenfunction, and performing this gives an eigenvalue(a number) and the original function. An eigenvalue can not include any other variables.
legendre polynomials
P0 (x) = 1 // P1 (x) = x // P2 (x) = 1/2 (3x^2 -1) where m = 0 // orthogonal to each other the solutions P(x) depend on m and l where l=1,2,3... and m= 0,+-1,+-2,....+-l. P(x) = Θ(θ) polynomials found in the legendre equation, which solves for Y(θ,φ) (rigid rotator wave functions)
ultraviolet catastrophe
Rayleigh-Keans law reproduces the experiment at low frequencies but diverges at high frequencies (short wavelength) as radiation enters the ultraviolet range// rayleigh-Jeans law about blackbody radiant energy density - first failure in CLASSICAL physics trying to explain quantum ideas
Rydberg constant
Rh = 109677.57 cm-1
classical limit
The large quantum-number limit described in the Correspondence Principle is the classical limit
Laplacian operator
The partial derivatives in schrodingers dimensional equation with x,y and z axis // ∇2
free-electron model
The particle in a box model can be applied to electrons moving freely in a molecule (AKA free electron model)
commutator
[A^,B^] = A^B^ - B^A^. <<< perform this to get the commutator (one of the hw probs). It shows if you can measure two "observables" simultaneously.
operator
a symbol that tells you to do something (a mathematical operation) to whatever (function, number, etc.) that follows the symbol. (uses the ^ symbol)
blackbody
an ideal body, which absorbs and emits all frequencies // ideal blackbodies do not exist - most absorb/emit all frequencies in a limited range of frequencies
series limit
as n increases the lines bunch up towards this // the limit of wavelengths/frequencies allowed for the transitions in the series (lyman limit is UV light / Balmer limit is Vis light / Bracket limit is IR light) // obtained for n2 -> ∞
wave-particle duality
light behaves as a wave in some experiments and as a stream of photons in others
line spectra
certain discrete frequencies from the emission spectra of atoms // simplest is the hydrogen atom
Rydberg formula
the generalized balmer formula when other series discovered in UV and IR // ν~ = 1/λ = Rh (1/n1^2 - 1/n2^2) where n1 < n2 therefore n1 is the lower level that the electron originated from
zeeman effect
degeneracy (2l+1) of each energy level is removed in magnetic field (becomes unequal energies due to diff m values)
Balmer formula
describes the lines in the hydrogen spectrum occurring in VIS and near UV regions // ν= 8.2202e12(1-4/n^2) Hz /// in wavenumbers -> ν~ = 109680 (1/2^2 - 1/n2^2) cm-1
quanta
discrete increments/units // energy quantization : an oscillator acquires energy in these discrete units
de Broglie wavelength
exhibited by a particle of mass, m, moving with a speed, v
odd function
f(x) = -f(-x)
even function
f(x) = f(-x)
force constant (k) ***
force required to produce unit extensions or compression. Used in the harmonic oscillator
normalizable
function is this if ∫Ψ*(x)Ψ(x)dx = A not = 1 because they can be normalized if they are divided by √A
angular momentum
given by ehrenfest rule // denoted by L // L =mevr = n(h/2π) = nħ // is quantized -> v= nh/2πmer
Heisenberg Uncertainty principle
if we wish to locate an electron within a region Δx there will be uncertainty in the momentum of the electron Δp that is greater than h/4π // ΔxΔp>= h/4π or ħ/2 OR uncertainty in energy and time ΔEΔt>= h/4π or ħ/2
rotational constant (B)
if you know B, you can solve for r, and then solve for I
azimuthal/secondary/orbital quantum number
l // describes the type of sub shell(or shape) // l = values between 0 and n-1 // assist in determining radial function // completely determines the magnitude of angular momentum of the electron about the nucleus (proton) l=0 s l=1 p l=2 d l=3 f
magnetic quantum number
m // can take (2l+1) values : m=0 to +- l // describes orientation or type of orbitals, and assists in determining actual orbital // completely determines the z component of the angular momentum // determines the energy of hydrogen atom in a magnetic field
threshold frequency
minimum frequency of light required to eject and electron // below which no electrons eject // Vo // above Vo the KE of the electrons varies linearly with frequency
principle quantum number
n // describes both the shell or level and sub shell // n=1,2,3,... // assist in determining radial function
total number of nodes
n-1
Lyman series
n1 = 1 n2= 2,3,... // UV range ~100-400 nm higher E difference
Bracket series
n1= 4 n2=5,6... // IR rang ~700 nm - 1 mm smaller E difference
Balmer series
n1=2 n2=3,4... // Vis range ~400-700 nm
tunneling
nonexistent in classical mechanics - a QM property that the wave function is non-zero in classically forbidden zones
angular node
number of nodes in the angular part is given by l // when Y (θ,φ) = 0
photoelectrons
observed emitted electrons from the surface of solids upon incidence of light having suitable wavelengths depends on frequency NOT intensity of light but number of electrons emitted is proportional to intensity of light
stationary state or orbit
observed spectrum is due to transitions from one allowed energy state to another one // spectral "terms" from ritz rule // both assumed these to exist so the electron is not accelerate toward the nucleus
nodal plane
plane that crosses through angular part at which no electrons are present
associated Laguerre polynomials
polynomials found at the end of the radial part of wave function ( back part of equations whole right section) polynomials are dependent on n and l
Correspondence Principle
quantum mechanics results and classical mechanics results tend to agree in the limit of large quantum numbers
blackbody radiation
radiation emitted by the blackbody
rigid rotator
refers to the QM treatment of rotational motion - a model for a rotating diatomic molecule
Particle-in-a-box
refers to the quantum mechanical treatment of translational motion. Solving Schrödingers equations obtains the wavefunctions and the allowed energies for the particle. Energy levels and energy separation b/w the energy levels increases as the size of the box or mass decreases
harmonic oscillator
refers to the quantum mechanical treatment of vibrational motion. the total energy is conserved, it is transferred between K and V. It is a model for vibrations in diatomic molecules
Wein displacement law
relationship between the wavelength at maximum intensity and the temperature; λmaxT=constant = 2.8979e-3 mK
degeneracy
represents the property of two or more eigenfunctions having the same eigenvalue. (AKA levels contain multiple wave functions with the same energy). = 2J+1
photon
small packets (quanta) of light of radiation (blackbody) // proposed by einstein E = hν
IR spectrum
the QM harmonic-oscillator model accounts for the IR spectrum of a diatomic molecule
Planck constant
the constant h in the planck distribution law for blackbody radiation // law reproducible for all frequencies if h=6.626e-34 Js and temps // explains the constant in weins law
complete set
the eigenfunctions of QM operators form a complete set
orthogonal
the eigenfunctions of the QM operators are orthogonal. If you know something about one, you know nothing about the other. If the eigenvalues of the wave functions are real, then the integral of product of the two functions = 0.
fundamental vibrational frequency
the existence of only one frequency in the spectrum of a diatomic molecule - harmonic oscillator predicts this
microwave spectroscopy
the frequency of transitions in the rigid-rotator is in the microwave range of electromagnetic radiation (10^10 - 10^11 Hz)
complex conjugate
the function Ψ. It is obtained by replacing i with -i in the expression of the wavefunction Ψ. The produce ΨΨ becomes real. Ψ*(x)Ψ(x)dx is the probability that the particle to be located between x and x+dx.
Schrödinger equation
the is the fundamental equation of QM for finding the wave function of a particle. Based on the idea that if the matter possesses wavelike properties there must be a wave equation that governs them. It cannot be demonstrated, only seen as a fundamental postulate. Time independent // for 1-D 2-D and 3-D
Bohr frequency condition
the lines in the spectrum were a result of two allowed stationary states (terms) and planck-einstein equation E=hν holds with the energy being equal to energy difference between two states, n1 and n2 (energy difference is same thing as the light absorbed/emitted hν)
most probable value of r
the maximum probability // the first bohr radius is equal to the most probable radius for the 1 s orbital // the average vale of r for 1s is = 3/2 ao (bohr radius)
work function
the minimum energy required to remove an electron from the surface of a particular metal // denoted Φ // KE of the ejected electrons is the difference between the incidence photon hν and the min energy to eject // KE = 1/2 mv^2 = hν - Φ = hν - hνo where Φ=hνo where νo is the threshold frequency // usually eV
reduced mass (μ)
the movement of a two-body system can be reduced to the movement of a one-body system with a mass equal to the "reduced mass" of the two-body system. Shown by μ on the equation sheet, but make sure to multiply by the conversion from amu to kg!!
zero-point energy
the residual energy of the harmonic oscillator, it is different than zero, and it is obtained for the vibrational quantum number n=0
spherical harmonic functions
the rigid-rotator wave functions
first Bohr radius
the smallest radius, obtained for n=1 // ao = 5.292 e-11 m = 0.5292 A = 1 bohr
boundary surface
the surface 9of equal density) that contains 90% of electron density
selection-rule
the transitions between various levels in harmonic oscillator follow this rule - that there can only be transitions between adjacent states
angular momentum magnitude
the value of L by solving from the eigenvalue of the angular momentum operator. The eigenvalue itself give the square of it
atomic orbitals
these are given by solving the schrodinger equation for the hydrogen atom, one obtains the allowed energies (eigenvalues) and the wave functions of the electron (eigenfunctions)
term
transitions are said to take place between two terms // another name for energy level // Ritz combination rule // T1 = Rh/n1^2 and T2 = Rh/n2^2 and that ν~= T1-T2 essentially the breakdown of rydberg eqt //
variance/standard deviation
variance is a statistical mechanics quantity, the variance of experiments becomes zero bc the standard deviation is zero, so the only values observed are the An values (the eigenvalues)
normalized
wave functions that satisfy the condition of ∫Ψ*(x)Ψ(x)dx = 1
operand
what follows the operator
associated legendre polynomials
when m does not equal 0 there is a structured equation that you fill in based on m and l values // P ml (x) // normalizable
ground state
when n=1 // E1 = -13.6 eV = -1312 kJ/mol = -109690 cm-1
excited states
when n=2,3,... // E2 = -3.4 eV = -328 kJ/mol E3 = -1.5 eV = -134 kJ/mol
