Three Pictures

7/27/2019 Three Pictures

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Outline of the Talk

Brief review of (or introduction to) quantummechanics.

3 different viewpoints on calculation.

Schrdinger, Heisenberg, Dirac

A worked-out example calculation.

Other interpretations & methods.


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The Wave Function

A particle or system is described by its wave function.

The wave function lives in a well-defined space (typically aHilbert space) described by some set of basis vectors.

The natural language for our discussion is finite-dimensionallinear algebra, although this is all valid for other spaces.


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The Wave Function

The wave function satisfies Schrdingers differential equation,which governs the dynamics of the system in time.

The Hamiltonian of the system, , is the operator whichdescribes the total energy of the quantum system.


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Wave Function Example

Example: An atom with two accessible energetic space.

The space may be completely described by two basisvectors.

All possible states of the atom may be expressed aslinear combinations of these basis vectors.


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Wave Function Example

, generally.


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Quantum Statistics

The Copenhagen interpretation of quantum mechanics tells uscomplex square of the wave function gives theprobability density

function (PDF) of a quantum system.

For the complex square to be meaningful statistically, we need

the probabilities to sum to 1.


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Commutators

In general, two operators do not commute in quantummechanics.

The commutator measures the difference between the twocomposite operations.

Example: Canonical Commutation Relation


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Quantum Statistics

The probability of an observation is found by computing matrixelements.

If a quantum system is in a superposition of states, we findprobabilities through inner products.

This is just Fourier theory!


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Early Questions in Quantum Mechanics

Question of Interpretation What does quantum mechanics mean?

Inherently statistical

Question of Execution How do we do calculations?

What is a state? How does a state change in time?

3 Different Answers


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The Three Pictures of Quantum Mechanics

Schrdinger Heisenberg Dirac


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Schrdinger

Quantum systems are regarded aswave functions which solve theSchrdinger equation.

Observables are represented byHermitian operators which act onthe wave function.

In the Schrdinger picture, the

operators stay fixed while theSchrdinger equation changes thebasis with time.


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Heisenberg

In the Heisenberg picture, it is theoperators which change in time whilethe basis of the space remains fixed.

Heisenbergs matrix mechanicsactually came before Schrdingers

wave mechanics but were toomathematically different to catch on.

A fixed basis is, in some ways, moremathematically pleasing. Thisformulation also generalizes moreeasily to relativity - it is the nearestanalog to classical physics.


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Dirac

In the Dirac (or, interaction)picture, both the basis and theoperators carry time-dependence.

The interaction picture allows foroperators to act on the state vectorat different times and forms the

basis for quantum field theory andmany other newer methods.


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The Schrdinger Picture

In the Schrdinger picture, the operators are constant while thebasis changes is time via the Schrdinger equation.

The differential equation leads to an expression for the wave function.


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A quantum operator as the argument of the exponentialfunction is defined in terms of its power series expansion.

This is how the states pick up their time-dependence.


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The Heisenberg Picture

In the Heisenberg picture, the basis does not change with time.This is accomplished by adding a term to the Schrdingerstates to eliminate the time-dependence.

The quantum operators, however, do change with time.


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We may define operators in the Heisenberg picture viaexpectation values.

Operators in the Heisenberg picture, therefore, pick up time-dependence through unitary transformations.


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We may ascertain the operators time-dependence throughdifferentiation.

The operators are thus governed by a differential equationknown as Heisenbergs equation.


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The Dirac Picture

The Dirac picture is a sort of intermediary between theSchrdinger picture and the Heisenberg picture as both thequantum states and the operators carry time dependence.

Consider some Hamiltonian in the Schrdinger picture

containing both a free term and an interaction term.

It is especially useful for problems including explicitly time-dependent interaction terms in the Hamiltonian.


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The Dirac Picture

In the interaction picture, state vectors are again defined astransformations of the Schrdinger states.

These state vectors are transformed only by the free part ofthe Hamiltonian.

The Dirac operators are transformed similarly to the Heisenbergo erators.


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The Dirac Picture

Consider the interaction picture counterparts to theSchrdinger Hamiltonian operators.

The interacting term of the Schrdinger Hamiltonian is definedsimilarl .

This follows from commuting with itself in the seriesexpansion of the exponential.


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The Dirac Picture

States in the interaction picture evolve in time similarly toHeisenberg states but with a twist.


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The Dirac Picture

Therefore, the state vectors in the interaction picture evolve intime according to the interaction term only.

It can be easily shown through differentiation that operators inthe interaction picture evolve in time according only to the free

Hamiltonian.


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Matrix Elements

Schrdinger Picture

Heisenberg Picture

Dirac Picture


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Time-Dependent Commutators

Canonical Commutation Relation

Schrdinger Picture Representation


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Heisenberg & Dirac Pictures (No Interaction)

1-D Harmonic Oscillator

Operator time-dependence


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Now have time-dependent commutators.


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The Jaynes-Cummings Hamiltonian

Describes an atom in an electromagnetic field.

Only two accessible energy levels.

Perfectly reflective cavity, no photon loss.

Assume zero detuning, i.e. .


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Energy contained in the free field.


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Energy of the atomic transitions between

energy levels.


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Interaction energy of the field and the atom

itself.


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Split the Hamiltonian into two commuting terms.


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Initial state is a superposition of photon states.

We assume the atom to be in its excited state.


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The free Hamiltonian adds only a phase factor!

Now expand the interacting term in a power series...


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Cosine plus Sine!


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The fully time-dependent state may now be written.


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We may calculate, for example, the average atomic inversion.


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Expected value for the atomic inversion for mean photonnumber N = 25 (light blue) and N = 100 (dark blue).


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The Dirac Picture

Making the unitary transformation

leads to wave function dependence only on the interaction term.

Make the same series expansion as before (no phase factors).


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The Dirac Picture

We need to calculate the time-dependence of the inversionoperator to obtain an expectation value.

Everything commutes no time-dependence.


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The Dirac Picture

The expectation value of the inversion operator gives the sameresult, as required.


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Summary: Three Pictures & Other Forms

Schrdingers Wave Mechanics

Heisenbergs Matrix Mechanics

Diracs Canonical Quantization

Methods of Calculation

Feynmans Path Integrals / Sum Over Histories

Constructive Field Theory

QFT in Curved Spacetime

Many-Worlds Interpretation

Quantum De-coherence

Other Notable Interpretations


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http://en.wikipedia.org/wiki/Interaction_picturehttp://en.wikipedia.org/wiki/Schrhttp://en.wikipedia.org/wiki/Schrhttp://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanicshttp://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanicshttp://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanicshttp://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanicshttp://en.wikipedia.org/wiki/Heisenberg_picturehttp://en.wikipedia.org/wiki/Heisenberg_picturehttp://en.wikipedia.org/wiki/Interaction_picture

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