quantum mechanics note, Summaries of Physics

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Three Pictures of

Quantum Mechanics

Thomas R. Shafer

April 17, 2009

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.

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.

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.

Wave Function Example , generally.

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.

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!

Three Different Ways to Calculate

The Three Pictures of Quantum Mechanics Schrödinger Heisenberg Dirac

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.

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.

The Schrödinger Picture ˆ O != ˆ O (t)

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.

The Heisenberg Picture