three pictures of quantum mechanics - universityuncw.edu/phy/documents/shafer_09.pdf · the three...
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Three Pictures ofQuantum Mechanics
Thomas R. ShaferApril 17, 2009
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Outline of the Talk
• Brief review of (or introduction to) quantum mechanics.
• 3 different viewpoints on calculation.
• Schrödinger, Heisenberg, Dirac
• A worked-out example calculation.
• Other interpretations & methods.
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Setting the Stage:Quantum Mechanics in Five Minutes
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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 a Hilbert space) described by some set of basis vectors.
• The natural language for our discussion is finite-dimensional linear algebra, although this is all valid for other spaces.
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The Wave Function
• The wave function satisfies Schrödinger’s differential equation, which governs the dynamics of the system in time.
• The Hamiltonian of the system, , is the operator which describes 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 basis vectors.
All possible states of the atom may be expressed as linear 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 us complex square of the wave function gives the probability 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 quantum mechanics.
• The commutator measures the difference between the two composite operations.
• Example: Canonical Commutation Relation
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Quantum Statistics
• The probability of an observation is found by computing matrix elements.
• If a quantum system is in a superposition of states, we find probabilities through inner products.
This is just Fourier theory!
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Three Different Waysto Calculate
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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
Schrödinger Heisenberg Dirac
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The Three Pictures of Quantum Mechanics
Schrödinger
• Quantum systems are regarded as wave functions which solve the Schrödinger equation.
• Observables are represented by Hermitian operators which act on the wave function.
• In the Schrödinger picture, the operators stay fixed while the Schrödinger equation changes the basis with time.
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The Three Pictures of Quantum Mechanics
Heisenberg
• In the Heisenberg picture, it is the operators which change in time while the basis of the space remains fixed.
• Heisenberg’s matrix mechanics actually came before Schrödinger’s wave mechanics but were too mathematically different to catch on.
• A fixed basis is, in some ways, more mathematically pleasing. This formulation also generalizes more easily to relativity - it is the nearest analog to classical physics.
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The Three Pictures of Quantum Mechanics
Dirac
• In the Dirac (or, interaction) picture, both the basis and the operators carry time-dependence.
• The interaction picture allows for operators to act on the state vector at different times and forms the basis for quantum field theory and many other newer methods.
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The Schrödinger Picture
O != O (t)
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The Schrödinger Picture
• In the Schrödinger picture, the operators are constant while the basis changes is time via the Schrödinger equation.
• The differential equation leads to an expression for the wave function.
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The Schrödinger Picture
• A quantum operator as the argument of the exponential function 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
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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 Schrödinger states to eliminate the time-dependence.
• The quantum operators, however, do change with time.
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The Heisenberg Picture
• We may define operators in the Heisenberg picture via expectation values.
• Operators in the Heisenberg picture, therefore, pick up time-dependence through unitary transformations.
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The Heisenberg Picture
• We may ascertain the operators’ time-dependence through differentiation.
• The operators are thus governed by a differential equation known as Heisenberg’s equation.
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The Dirac (Interaction) Picture
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The Dirac Picture
• The Dirac picture is a sort of intermediary between the Schrödinger picture and the Heisenberg picture as both the quantum states and the operators carry time dependence.
• Consider some Hamiltonian in the Schrödinger 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 as transformations of the Schrödinger states.
These state vectors are transformed only by the free part of the Hamiltonian.
• The Dirac operators are transformed similarly to the Heisenberg operators.
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The Dirac Picture
• Consider the interaction picture counterparts to the Schrödinger Hamiltonian operators.
• The interacting term of the Schrödinger Hamiltonian is defined similarly.
This follows from commuting with itself in the series expansion of the exponential.
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The Dirac Picture
• States in the interaction picture evolve in time similarly to Heisenberg states… but with a twist.
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The Dirac Picture
• Therefore, the state vectors in the interaction picture evolve in time according to the interaction term only.
• It can be easily shown through differentiation that operators in the interaction picture evolve in time according only to the free Hamiltonian.
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Matrix Elements
• Schrödinger Picture
• Heisenberg Picture
• Dirac Picture
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Demonstration:The Canonical Commutation Relation
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Time-Dependent Commutators
• Canonical Commutation Relation
• Schrödinger Picture Representation
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Time-Dependent Commutators
• Heisenberg & Dirac Pictures (No Interaction)
• 1-D Harmonic Oscillator
• Operator time-dependence
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Time-Dependent Commutators
• Now have time-dependent commutators.
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A Worked Example:The Jaynes-Cummings Hamiltonian
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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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The Jaynes-Cummings Hamiltonian
• Energy contained in the free field.
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The Jaynes-Cummings Hamiltonian
• Energy of the atomic transitions betweenenergy levels.
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The Jaynes-Cummings Hamiltonian
• Interaction energy of the field and the atomitself.
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The Jaynes-Cummings Hamiltonian
• Split the Hamiltonian into two commuting terms.
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The Jaynes-Cummings Hamiltonian
• Initial state is a superposition of photon states.
• We assume the atom to be in its excited state.
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The Schrödinger Picture
• The free Hamiltonian adds only a phase factor!
• Now expand the interacting term in a power series...
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The Schrödinger Picture
• Cosine plus Sine!
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The Schrödinger Picture
• The fully time-dependent state may now be written.
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The Schrödinger Picture
• We may calculate, for example, the average atomic inversion.
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The Schrödinger Picture
• Expected value for the atomic inversion for mean photon number 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 inversion operator 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 same result, as required.
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Other Interpretations & Forms of Calculation in Quantum Mechanics
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Summary: Three Pictures & Other Forms
• Schrödinger’s Wave Mechanics
• Heisenberg’s Matrix Mechanics
• Dirac’s Canonical Quantization
• Methods of Calculation
• Feynman’s 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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Thank You!
References
• E. Jaynes and F. W. Cummings, Proceedings of the IEEE 51,
89 (1963)
• D. J. Gri!ths, Introduction to Quantum Mechanics, 2nd ed.
• http://en.wikipedia.org/wiki/Heisenberg_picture
• http://en.wikipedia.org/wiki/
Mathematical_formulation_of_quantum_mechanics
• C. C. Gerry and P. L. Knight, Introductory Quantum Optics
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed.
• http://en.wikipedia.org/wiki/Schrödinger_picture
• http://en.wikipedia.org/wiki/Interaction_picture
• T. Shafer, Collapse and Revival in the Jaynes-Cummings-Paul
Model