Ch 4!!! Fundamentals of Quantum Mechanics Flashcards
two observations that scienctists in the end of 19th century just couldn’t understand
- atoms, when inserted in flame, emit light of certain (and not any) colour
2.heated objects (such as a frying pan) emit light of particular colours, these change with temperatures
how does quantum theory help us
most complete theory we have to describe the behaviour of electrons, in turn explains how many biological, physical, and chemical processes work. ex, electronic devices rely on quantum mechanical nature of electrons
diffraction
the bending of waves by an obstacle
interference
property of all waves, waves interfere with each other and themselves either constructively or destructively
diffraction generally produces concentric circles
concentric circles
circles with a common center, in the case of waves this would be the origin
how does diffraction occur
as each wave/electron/particle hits atom in crystal lattice, wave splits and it’s components interfere with each other NOT with other electrons
momentum
mass x velocity
wavelength
distance between nearest points repeating pattern (trough to trough)
de broglie’s equation
fancy upside down y h thing = h/p. and p = mv
when fancy thing = wavelength (m)
p = momentum of the particle. kg m s^-1
m = mass of particle in kg
v = velocity of particle in m s^-1. or m/s
h = planck’s constant (6.626 x 10^-34 Js
why can we detect wavelike behaviour in electrons etc, not larger particles like tennis balls??
due to the value of h being so small, if you were to divide it by the velocity of a large object you would get a wavelength immensely smaller than the volume of the object being observed. Because of this larger particles do not exhibit observable wavelike properties
how can you maximize diffraction of particles
wavelength should be the same as spacing in lattice etc they will be passing through. If smaller than, diffraction will not occur and if greater than, each particle will be diffracted by many atoms etc and the diffraction pattern will be blurred
travelling waves
repeating and periodic disturbances that travel from one location to another
what do waves transport??
ENERGY not matter, energy that disturbs still medium. Electromagnetic ration transmits energy through oscillating electric and magnetic fields
Definition of a Transverse Wave, how can they be represented??
wave which oscillates in direction perpendicular to direction of motion
Amplitude, Wavelength, Frequency
the waves intensity and energy are proportional to the square of it’s amplitude
waves intensity and energy proportional to ___
the square of it’s amplitude
amplitude
difference between midpoint and crest or trough
wavelength
length of one complete cycle, crest to crest or trough to trough
frequency
complete cycles the wave can transmit per unit time
speed of wave
velocity, unit of distance divided by unit of time
velocity in m s^-1 = frequency (italic v, in s^-1 or Hertz) x wavelength (fancy h y in meters)
EQUATIONS TO KNOW
velocity (v) = frequency (italic v) x wavelength
wavelength in m= h(planck’s constant) Js / momentum of a particle in kg m s^-1
Kinetic energy = 1/2 m v^2
momentum in kg m s^-1 = mass of particle (kg) x velocity of particle in m s^-1
E = h^2 n^2 / (8mL^2)
change in energy = h^2 / 8mL^2 x (Ef^2 - Einitial^2)
what happens to the wavelength of a particle when velocity decreases
as velocity decreases, momentum decreases and thus wavelength increases
basic sin function of a wave
f(x) = Asin (2pix/wavelength)
A is a constant, amplitude
standing wave
bound on either side (length L), appear to be still and do not carry energy from one place to another
how can standing wave be formed
travelling waves of equal amplitude and frequency but in opposite directions interfere with each other
nodes
points where amplitude = 0, generally where waves deconstructively interfere with each other
where is particle on a wave??
it can exist simultaneously everywhere, position can only be predicted - PROBABILITY
heisenberg uncertainty principle
we cannot know precisely (with zero uncertainty) both the position and momentum of a microscopic particle at the same time
particle at rest, velocity momemtum annd KE?
all equal to zero
probability distribution of particle in a box
uniform across length of box, and equal to 1/L
remember when predicting average momentum
momentum is a VECTOR both magnitude and direction
psi (fancy wierd greek y)
WAVEFUNCTION - all info that can possibly be obtained from a quantum system in a given state
probability distribution of a wave??
find by taking the square of the wavefunction
schrodinger equation
not even going to try to type, google
what does potential energy describe
ex hydrogen atom n, electron moving in coulomb field of proton, v is PE of the electron from coulombic force between nucleus and electron
PE (V) will depend on the problem
schrodinger for particle in a box
-h^2/ 8pi^2m x d^2/dx^2 psi = Epsi
d^2/dx^2 psi just means the derivative of the wavefunction
wavefunction for quantum particle in a box
psi = Asin (kx)
COMPLETE
psi = √(2/L) sin ((npix)/L)
n
principle quantum number, must equal a positive integer number
why can’t n be a fraction?
with quantum particle in a box function, psi must equal 0 at L, sin (kL) = 0, this only happens at sin(npi), therefore n can only be a positive integer
value of amplitude quantum particle in a box
√(2/L)
where is the maximum of probability graph?
if amplitute psi = √2/L amplitude of wavefunction equals square of this
ex if L = 1 amplitude of prob graph is 2
how do you known # of nodes in quantum particle in a box??
n - 1
equation for energy of particle in a box
E = h^2 n^2 / (8mL2)
ground state
E1 = h^2 / 8mL^2.
LOWEST possible energy
always greater than 0
zero point energy
difference between the ground state energy and zero
excited state
any state n>1, so 2, 3, to infinity
n=2 is first excited state
how can particles change their energy levels
absorbing or emitting energy, ex recieve energy from light or from collisions
equation change in energy level
change in E = E final - E initial
E final energy of the system in final state (nfinal)
E initial, initial energy original n
change in E = h^2/(8mL^2) x (nfinal^2 -ninitial^2
what does change in energy level mean?
if positive, system absorbed energy
if negative, system emit energy
can particle gain lose completely random amounts of energy??
no, quantized, quantum system never observed between states, can only gain or lose exact amounts of energy
determining n, wavelength, function (psi), and probability distribution function given # of nodes
n = # of nodes + 1
wavelength = 2L/n
function = √(2/L) sin ((n pi x)/L)
probability distribution function = 2/l sin^2((n pi x) /L)
how to calculate any excited energy state
calculate ground state using h^2 / 8mL^2
multiply this by n^2
quantized
can only take on certain values
