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Quantum Mechanics
Quantum Mechanics
Quantum mechanics
- wave-particle duality- explanation of energy states in complex atoms and molecules- the intensities of spectral lines- explains a wealth of other phenomena.- accepted as being the fundamental theory underlying all physical processes- “entanglement”
Quantum Mechanics – A New Theory
Bohr’s quantum theory- theory for transitions in atoms – radiating atoms- stability of the atomNot: complex atoms, line intensities, molecules
Quantum Mechanics
mvp =
Dealing with the wave-particle duality:
Holds both for light waves and matter waves
Light:
Matter:
cEp /=
.
For the momentum of the particle
What about the wave character ?
The Wave Function and Its Interpretation; the Double-Slit Experiment
2Ψ
Interference for the waveamplitudes:
- E field for electromagnetism
−Ψ for the amplitude of a matter wave
Matter behaves like a wave
Interpretation of amplitudes
Light intensity
Matter
NEI ∝∝ 2
Young’s Double-Slit Experiment
Can be meausured for;Light and for electrons
Both entities can be wavesand particles
Even at extremely lowintensities: single-particleinterference ?
Indeterminism:You cannot predict where oneparticle ends up
Born interpretation of quantum mechanics; the wave function
( )2,trrΨ Represents the probability to find a particleAt a location r at a time t
The probability densityThe probability distribution
The Nobel Prize in Physics 1954"for his fundamental research in
quantum mechanics, especially for his statistical interpretation of the wavefunction"
Max Born
Fundamental limits to measurement –inherently.
- wave-particle duality
- interaction between equipment and the object being observed
The Heisenberg Uncertainty Principle
Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing so it will transfer some of its momentum to the electron.
This is called the Heisenberg uncertainty principle, in a more refinedcalculation:
It tells us that the position and momentum cannot simultaneously be measured with precision.
The Heisenberg Uncertainty Principle
λ≈Δx
λhpx ≈Δ
A rough estimate:
If light is used to observe a particle, the uncertainty in position is
If the object is detected by a single photon; momentum is transferredto object;
This yields: ( )( ) hpx x ≈ΔΔ
Heisenberg uncertainty for time and energy (ignore the 2π):
This says that if an energy state only lasts for a limited time,its energy will be uncertain. It also says that conservation of energy can be violated if the time is short enough.
The Heisenberg Uncertainty Principle
ccxt // λ≈Δ≈Δλ≈Δx
Consequences for the width of a short-lived quantum state
Note the relation with waves in optics and electronics
Location
Energy
Time
λ/hcE ≈Δ
The Heisenberg Uncertainty PrincipleCalculations
An electron moves in a straight line with a constant speed v = 1.10 x 106 m/s, which has been measured to a precision of 0.10%. What is the maximum precision with which its position could be simultaneously measured?
What is the uncertainty in position, imposed by the uncertainty principle, on a 150-g baseball thrown at (93 ± 2) mph = (42 ± 1) m/s?
Quantum mechanics is meant for electrons, not for baseballs
The Heisenberg Uncertainty Principle
The J/ψ meson, discovered in 1974, was measured to have an average mass of 3100 MeV/c2 (note the use of energy units since E = mc2) and an intrinsic width of 63 keV/c2. By this we mean that the masses of different J/ψ mesons were actually measured to be slightly different from one another. This mass “width” is related to the very short lifetime of the J/ψbefore it decays into other particles. Estimate its lifetime using the uncertainty principle.
The world of Newtonian mechanics is a deterministic one. If you know the forces on an object and its initial velocity, you can predict where it will go.
Quantum mechanics is very different – you can predict what masses of electrons will do, but have no idea what any individual one will do.
Philosophic Implications; Probability versus Determinism
Einstein-Podolsky-Rosen paradoxUse indeterminism in a quantum computer
The Schrödinger Equation in OneDimension—Time-Independent Form
The Schrödinger equation cannot be derived (??), just as Newton’s laws cannot. However, we know that it must describe a traveling wave, and that energy must be conserved.
Therefore, the wave function will take the form:
where
Since energy is conserved, we know:
This suggests a form for the Schrödinger equation, which experiment shows to be correct:
xip
∂∂
→h
Later: QM
The Schrödinger Equation in OneDimension—Time-Independent Form
Since the solution to the Schrödinger equation is supposed to represent a single particle, the total probability of findingthat particle anywhere in space should equal 1:
When this is true, the wave function is normalized.
Time-Dependent Schrödinger Equation
A more general form of the Schrödinger equation includes time dependence (still in one space dimension):
Derivation requires more rigorous methods of QM.
Ψ(x,t) is the wave function dependent on space-time coordinates.
The time-independent Schrödinger equation can be derived from it,using the method of “separation of variables”.
Note that this is a non-relativistic wave equation: Later a Lorentz-covariant formalism gives the DIRAC equation
“Separation of variables” - method
f
fdtdiU
dxd
m hh
=+−
ψ
ψψ2
22
2
fdtdifUf
dxd
mψψψ h
h=+− 2
22
2
ffdtdi λ=h
Assume the potential to be time-independent U(x) , and the trial solution
then
Move space coordinates to left and time to right; divide by Ψ
Left and right side must be independent,equal to a constant, say λ
Solution: ( ) h/tietf λ−=
Oscillating function of time, with frequencyh
λω =
Associated with energy: λω == hE ( ) h/iEtetf −=Hence:
Stationairy states in QM
For a QM problem with a time-independent potential U(x)
( ) h/)(, iEtextx −=Ψ ψ
The solutions are stationairy states:
( ) 2//**2 )()()(, xexextx iEtiEt ψψψ ==ΨΨ=Ψ − hh
The probalistic aspects do not vary with time !
Stationairy states may have a time-dependent phase
Free Particles; Plane Waves and Wave Packets
)()(2 2
22xE
dxxd
mψψ
=−h
0)(2)(22
2=+ xmE
dxxd ψψ
h
kxx sin)( =ψ
Free particle: no force, so U = 0. The Schrödinger equation becomes the equation for a simple harmonic oscillator:
Independent solutions kxx cos)( =ψ
Also: ikxex =)(ψ ikxex −=)(ψ
Superposition principle of quantum mechanics:Any linear combination of solutions is also a solution
Free Particles; Plane Waves and Wave Packets
Hence solutions:
Since U = 0, Energy is only kinetic
where
Free Particles; Plane Waves and Wave Packets
The solution for a free particle is a plane wave, as shown in part (a) of the figure; more realistic is a wave packet, as shown in part (b). The wave packet has both a range of momenta and a finite uncertainty in width. (normalization problem)
How to describe a wave packet ?
(Famous) Particle in an Infinitely Deep SquareWell Potential (a Rigid Box)
Solution for the region between the walls
Requiring that ψ = 0 at x = 0 and x = gives B = 0 and k = nπ/ .
This means that the energy is limited to the values:
l
Particle in an Infinitely Deep SquareWell Potential (a Rigid Box)
plots of solutions with quantum number n
Zero-point energy = nonclassical
Particle in an Infinitely Deep SquareWell Potential (a Rigid Box)
dxxdx ∫∫ ⎟⎠⎞
⎜⎝⎛=Ψ
4/
0
24/
0
2 sin2 ll
ll
π
Probability of e- in ¼ of box.
Determine the probability of finding an electron in the left quarter of a rigid box—i.e., between one wall at x = 0 and position x = /4. Assume the electron is in the ground state.
l
Particle in an Infinitely Deep SquareWell Potential (a Rigid Box)
l
∫ Ψ=l
0
2)()( dxxxP
Most likely and average positions.
Two quantities that we often want to know are the most likely position of the particle and the average position of the particle. Consider the electron in the box of width = 1.00 x 10-10 m in the first excited state n = 2.
(a) What is its most likely position?
(b) What is its average position?
∫ Ψ=l
0
2)( dxxxx
Finite Potential Well
)()( ll IIIII ψψ =
)()( lldx
ddx
d IIIII ψψ=
)0()0( III ψψ =
A finite potential well has a potential of zero between x = 0 and x = , but outside that range the potential is a constant U0.
l
The potential outside the well is no longer zero; it falls off exponentially.
Solve in regions I, II, and IIIand use for boundary conditionsContinuity:
)0()0(dx
ddx
d III ψψ=
Bound states: E < E0Continuum states: E > E0
Finite Potential Well
( )202 2h
EUmG −=( ) 02
20
2
2=⎥⎦
⎤⎢⎣⎡ −
− ψψh
EUmdxd
GxGxIIII DeCe −+=,ψ
If E < U0
with
in the “forbidden regions”
General solution:
Region I 0<x hence 0=D and similarly for C
GxI Ce=ψ should match kxBkxAII cossin +=ψ
Finite value at 0=x exponentially decaying into the finite walls
Finite Potential WellThese graphs show the wave functions and probability distributions for the first three energy states.
Nonclassical effects
Partile can exist in the forbidden region
Finite Potential Well
( )02 UEmh
ph
−==λ
( ) 022
02
2=⎥⎦
⎤⎢⎣⎡ −
+ ψψh
UEmdxd
0222
2=⎥⎦
⎤⎢⎣⎡+ ψψh
mEdxd
If E > U0
In regions I and III
free particle condition
In region II
In both cases oscillating free partcilewave function:
I,III:
II: mEh
ph
2==λ
0
2
02
221 U
mpUmvE +=+=
Tunneling Through a Barrier
l>x
0<x
mEh
ph
2==λ
( ) 0220
2
2=⎥⎦
⎤⎢⎣⎡ −
− ψψh
EUmdxd
0222
2=⎥⎦
⎤⎢⎣⎡+ ψψh
mEdxd
Also in region
In region oscillating wave
mEh
ph
2==λWave with same wavelength
In the barrier:
GxGxb DeCe −+=ψ
Approximation: assume that the decayingfunction is dominant Gx
b De−=ψ
( )( )
( ) ll GGx
eD
Dex
xT 2
2
2
2
2
0−
−==
=
==ψ
ψTransmission:
Tunneling Through a Barrier
The probability that a particle tunnels through a barrier can be expressed as a transmission coefficient, T, and a reflection coefficient, R (where T + R = 1). If T is small,
The smaller E is with respect to U0, the smaller the probability that the particle will tunnel through the barrier.
Tunneling Through a Barrier
Alpha decay is a tunneling process; this is why alpha decay lifetimes are so variable.
Note:Exponential dependence
Region of binding by the “strong force”
Region of repulsion between positive charges
Tunneling Through a Barrier
Scanning tunneling microscopes image the surface of a material by moving so as to keep the tunneling current constant. In doing so, they map an image of the surface.
Gerd Binnig
"for their design of the scanning tunneling microscope"
HeinrichRohrer
Nobel 1986
