pchem exam2

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What are 2 key concepts that distinguish quantum from classical mechanics?

1. wave-particle duality

2. quantization of energy

When must you use quantum mechanics to study a problem?

atomic and subatomic levels where classical physics do not describe the system

What are 2 key concepts that distinguish quantum from classical mechanics?

1. wave-particle duality

2. quantization of energy

When must you use quantum mechanics to study a problem?

atomic and subatomic levels where classical physics do not describe the system

What must be true for a function to be an eigenfunction of an operator?

applying the operator to the function should yield the eigenfunction times an integer

What makes a function suitable as a wave function?

it must be single-valued, continuous, and have a continuous derivative

what makes 2 observables associated with 2 operators able to be determined simultaneously?

they must commute (applying them in a different order yields same result)

meaning of commutative operators?

they are in the same eigenspace

what is the quantum mechanics operator that describes a system’s total energy?

H- Hamiltonian

Blackbody radiation experiment

A blackbody emits radiation in all wavelengths dependent on only temperature. Classical theory predicts that a blackbody will emit an infinite amount of energy at all temperatures greater than absolute zero, which did not occur as emission peaked and then went down. Planck explained the dependence of spectral density on frequency for blackbody radiation by assuming that the energy radiated was quantized, as high frequency results differed greatly from classical.

photoelectric effect set-up

light hits a copper plate in a vacuum, shooting electrons out of it. the absorbed light energy is equal to the sum of the energy required to eject an electron and the kinetic energy of the emitted electrons because energy is conserved.

observations of the photoelectric effect

no electrons are emitted unless light is above a threshold frequency, kinetic energy of the electrons independent of the light energy but the amount of them is dependent.

photoelectric effect findings

Experiments to elucidate the photoelectric effect provided the first evidence of wave-particle duality for light. Light does not equally hit the copper, proving that it can be in little quanta

de Broglie relation

particles also behave like waves and have wavelengths based on their momentum

Schrodinger Equation

gives probability of finding a particle in a given time and position (basis for orbitals)

standing wave

time independent, position well-defined

relationship between eigenfunctions and operator

they must be orthogonal (perpendicular)

postulate 1

state of a quantum mechanic particle is completely specified by the wave function, which must be continuous (and its derivative) and single-valued, so that it is integratable for probability

postulate 2

for every measurable property of a system, there is a corresponding operator

postulate 3

for any single measurement of the observable that corresponds to operator A, the only values that will be measured are eigenvalues of the operator A

postulate 4

taking multiple measurements of identical systems should give the expected probability, but only for the 1st measurement of each one. the expected value is a weighted average of all possibilities

postulate 5

measurement interferes with a system, as the second measurement of a system is deterministic and not probabilistic. before measuring, all possibilities are superimposed (superposition state) and measurement collapses them all to one reality

drawing eigenfunctions for particle in a box

number of nodes=principal quantum number, draw nodes as n-1 in the box

what happens as n increases in 1D box?

probability becomes continuous and you reach classical mechanics because the probability is equal across the box

HOMO of pi system

number of pi bonds

LUMO of pi system

number of pi bonds + 1

when can observables be known simultaneously and exactly?

if multiple operators commute

why are conditions so different on a micro and macro scale?

larger particles have very small wavelengths and do not behave as much like waves, so there is less uncertainty associated with them

commutator

difference between applying AB and BA, 0 if the eigenfunctions commute. if they commute, they have a common set.