Fundamentals of Quantum Mechanics

Question 1

Young’s double slit experiment

Answer 1

Shows that light acts as a wave

Radial waves emitted by diffraction –> overlap & interfere

“a wave can be at the same point in space and time” –> CONSTRUCTIVE or DESTRUCTIVE interference -|- waves

Question 2

e- diffracted through 2 slits

Answer 2

same results as Young’s double slit experiment –> e- are waves too

quantum mechanically, the e- goes through both slits simultaneously BUT if you measure it, you’ll only find it going through one

Question 3

e- diffraction (e.g. x-rays)

Answer 3

light acts as a particle
e- diffracted at variety of FIXED angles –> particle behaviour

Question 4

Equation for photon’s E

Answer 4

E = hν

E = energy, J
ν (nu) = freq., Hz (s^-1)
h = Planck’s (proportionality) constant, Js (how many J per unit freq = per s^-1 –> cancels to s)

Question 5

Photoelectric effect

Answer 5

If photon E > work function of metal, photoelectric effect will occur = e- will be ejected from material

RED (low freq) = no e-
GREEN = e-
BLUE (high freq) = high KE e-

linear trend –> E is directly proportional to freq of light

Question 6

de Broglie wavelength

Answer 6

wavelength of light

𝛌 = h/p = h/mv

p = momentum, kg m s^-1
m = mass, kg
v = velocity, m s^-1

bigger mass = smaller 𝛌 (negligent) –> QM doesn’t matter for large objects (classical limit)
Bigger objects follow Newtonian mechanics
QM when 𝛌 of particles becomes similar to size of system

Question 7

ψ

Answer 7

wavefunction - holds all info about a system & is the solution to the Schrödinger equation

“the wavefunction is the function ψ(x) for which the s-eqn is satisfied)

Question 8

Schrodinger equation

Answer 8

KE + PE = total E x

(2nd derivative ψ)∝(V-E)ψ
- ignore constant
- all wrt x

THUS curvature of ψ∝(KE*ψ)

Question 9

definition: “an observable”, Ω

Answer 9

any measurable property of a physical system

Question 10

definition: “an operator”, Ω(hat)

Answer 10

any symbol that indicates an operation to be performed
e.g. √, d/dx, a hat over a letter

In QM, observables are defined by “mathematical” operations on ψ

Question 11

Hamiltonian operator

Answer 11

Ĥ = operator for the total E

allows s-eqn to be written in shorthand:
Ĥψ(x) = Eψ(x)
where Ĥ = - (ħ/2m)*(2drv) + V(x)

Question 12

ħ = ?

Answer 12

ħ = h/2π

Question 13

QM

Answer 13

A study of how matter exists when we think of matter as waves

Question 14

KE operator

Answer 14

Ê(kin) = -(ħ/2m)*(2drvψ)

Question 15

Ω(hat)ψ(x) = Ωψ(x)

Answer 15

In general, there exists an operator, Ω(hat), which acts on ψ(x) to yield an observable, Ω, multiplied by ψ(x)

Question 16

Operator for position in QM

Answer 16

x̂ = x

–> can write: x̂ψ(x) = xψ(x)

Question 17

S-eqn: curvature & magnitude

Answer 17

curv. ψ(x) of depends on whether V(x) > or < E

magnitude of curv. is defined by KE (V(x)-E)

Question 18

Normal regime (world)

Answer 18

V(x) < E

curvature∝(V-E)ψ
curvature = 2nd deriv (ψ)

so curv. ∝ -ψ
ψ(x) > 0 -ve curvature
ψ(x) < 0 +ve curvature

–> ψ(x) ALWAYS curves TOWARDS the x-axis

kinds of functions that naturally do this: sine & cosines

show: ψ(x) = sin(kx)
apply to free particle ψ: V(x)=0 AKA particle is moving under no external force
-(ħ/2m)(2drv_sin(kx)) = Esin(kx)
-(ħ/2m)(2drv_sin(kx)) = Esin(kx)

[INCOMPLETE]

d/dx (sin x) = cos x
d/dx (cos x) = -sin x