PChem Test 2

Classical Physics and Commuting

There is no limit to the amount of info (observables) that can be known about a system at a given instant

A system under consideration can be described completely. 

Quantum Mechanics and Commuting
two observables can be known simultaneously on ly if the outcome of measurements is independent of the othe order in whcih they are conducted.
Uncertainty Relation
Limits the degree to which obsevables of noncommuting operators ca be knownn simultaneously.
Similarity of Classical and Quantum – with Free Particle in 1D Box

Chapter 15

The quantum mechanical solution of the problem contains the same info as the classical particle problem, namely, motion iwth a constant velocity.

Another similarity is that both can take on all values of energy, becasue k is a continuous variable.. The quantum mechanical free particle has a continuous energy spectrum.

 

 

Commuting

  • The values of two different observalbe (a) and (b), which correspond to the operators (A) and (B), can be simultaneously determined only if the measruement porcess used does not change the state of the system.
  • Otherwise , the sytme on which the two measurements is carried out is not the same.
  • If two operators have a common set of eigenfunctions, – we say they commute
  • commutator – the operators in brackets.
  • if the value of commutator is not zero for an arbitrary function f(x), then the corresponding obeservables cannot be determined simultaneously and exactly. 

Act of Changing a measurement in Quantum mechanics

  • Communting / communtators describe why you can achieve different results based on the experiment in Quantum Mechanics.
  •  The act of the measurement changes the state of the systme unless the system wave function is an eigenfunction of the two diferent operators. – Therefore, this is a condition for being able to simultaneously know the observables corresponding to these operators.

Heisenberg Uncertainty principle
This principle quantifies the uncertainty in the position and memtum of a quantum mechanical particle that arises from the fact that [(x),(p)] Does not equal 0.
Normalization  
 – a plane wave cannot be normalized over an infinite interval, but it can be normalized over the finite interval (-L<=x<=L).
Commuting  
For operators to commute – the eigenfunction of two different operators must be the same function.
Intrinsic
a characteristic or property of some thing or action which is essential and specific to that thing or action, and which is wholly independent of any other object, action or consequence.
Intristic Angular Moment

SPIN

Electrons have this propertry – not a magnetic moment produced by orbiting electrons around nucleus.

This is the basis for the deflection in the Stern-Garlach experiment 

Stern Garlach Experiment     
designed to distinguish b/w quantum mechanical model of atom propsed by Bohr and the classical planetary models.
Classical and Quantum Mechanics – Particle in a Box

Chapter 15

Unlike in Classical Mechanics, where the energy spectrum is continuous and the particle is equally likely to be found anywhere in the box, the quantum mechanical particle in the box has a discrete energy spectruma dn has preferred postiions that depend on the quantum mechanical state

 

Free Particle Probability  

P(x)dx = dx/2L  [-L<=x<=L], for a plane wave

 NOTE: that the P(x)dx is independent of x.

This result states that the particle is equally likely to be anywhere in the interval, which is equivalent to saying tha tnothing is known about the position of the particle. 

Energy of Particle in a Box: limited by Boundary Condition [Psi(0)=Psi(a)=0]

The energy for the free particle without BC is continuous. 

 

The energy for the particle in the box can only take on discrete values, and we say that the enrgy of the particle in the box is quantized adn the integer (n) is a quantum number.

 

 

ZERO POINT ENERGY

The lowest allowed energy is greater than zero.

The particle has a nonzero minimum energy – ZPE. 

Comparison of Free Particle with/without Boundaries

A comparison of these two problems reeals that quantization entered through the onfinement of the particle.

Because the particle is confined to the box, the amplitude of all laloowed wave functions must be zero everywhere outside the box.

NOTE: if a approaches infinity, the oncinement condition is removed and the energy = 0 and is continuous as i the nonbounded condition.

 En increases in jumps rather than in a continuous fashion as in classical mechanics. – this helps imagine ZPE.

 

Wave Function and Eigenfunction  

Wave Function completely describes every system in quantum mechanics

it is any mathematically well-behaved function that satisfies the boundary conditions and that can be normalized to allow a meaning ful definiton of probability.

Eigenfunction must satisfy the above as well as one more criterion.  A wave function  is an eigen function of an operator A’ only it it satisfies the relationship A’psi(x) = (a)psi(x) 

Two Major Differences between teh finite and the infinite depth box

1. the potential has only a finite njumber of bound states, which are the allowed energies. The number of bound energies depends on m, a and Vnot.

2. The amplitude of the wave function does not go to zero at the edge of the box in the finite depth box. 

 

 

Finite Box

The wavefunction decays exponentially once it passes through the edged wall.

 The wave function falls off most rapidly with distance for the most strongly bound state (Vo>>E) in the the potential – and – falls off most slowly for th eleast strongly bound state in the potential (Vo~E).

  • Strongly bound state (1,2 small) correspond to core electrons
  • Weakly boound levels (4,5 large) corresond to valence electrons

 

Pi Electrons in Conjugated Molecules – Treated like particle in box

If the electrons are delocalized as in an organic molecule with a pi-bonded network, the maximum in the absoprtion spectrum shifts from the UV intot he visible range.

The greater the delocalization, – longer a – greater wavelength – lowered Energy, which produced more absorption towards the red end of the spectrum — it is a "red shift". – This is based on the lowered energy ROYGBIV 

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