Quantum Mechanics

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Quantum mechanics is a theory that describes the interactions of all particles and systems. It underlies all physical phenomena, including scattering.


Wavefunction

A quantum system is completely specified by its Wave Function:

Integral Notation Dirac Notation
     

The wavefunction is typically normalized:

Integral Notation Dirac Notation
     

The distribution of the particle described by is given by:

Integral Notation Dirac Notation
     

In the Copenhagen Interpretation, is the probability of finding the particle at location . In Universal Wave Function interpretations (e.g. MWI), can be thought of as the spatial distribution of the particle. The wavefunction contains all the information one can know about a system. It can thus be thought of as 'being' the particle/system in question. However, the wavefunction can be described in an infinite number of different ways. That is, there is not a unique basis for describing the wavefunction. So, for instance, one can describe the wavefunction using position-space or momentum-space:

These representations can be inter-related (c.f. Fourier transform):

State

Note that the wavefunction describes the state of the system; there are various choices of basis one can use as an expansion.

An operator defines a particular convenient basis: one can always expand using the basis defined by an operator, in which case the above are the eigenvectors (or eigenstates) of that basis. This can also be viewed as a vector in the Hilbert space. The Dirac notation (bra-ket notation) is useful in this regard. A particular state is a (column) vector:

Which is a 'ket'. We define a 'bra' (the 'final state') as a (row) vector:

And note that the 'bra' is the conjuagte transpose of the 'ket':

Wave packet

TBD

Heisenberg Indeterminacy Relations

(Also known as Heisenberg Uncertainty Principle.)

Superposition

If and are both allowed states for a given system, then the following state is also allowed:

This leads to a notable consequence:

Notice that the final terms represent 'interference' between the two constituent states. This interference has no classical analogue; it is a quantum effect. Thus a superposition is not merely a 'joining' of the two states (e.g. "the particle can be in state 1 or state 2"), but a truly coherent interference between the two states. The superposition may be more generally written as:

Integral Notation Dirac Notation
     

The distribution of the particle described by is given by:

Integral Notation Dirac Notation
     

Operators

Observables in QM appears as operators ().

Examples: TBD.

Measurement

The transition of the wavefunction into state can be thought of as:

     

When acting on a wavefunction with operator the probability that the wavefunction ends up in state is given by:

     

The solutions take the form of an eigenvalue problem:

The allowed solutions of the equation, for operator , involve an eigenstate with associated eigenvalue . A measurement on a quantum system can be thought of as driving the wavefunction into an eigenstate defined by the operator; the value of the associated observable is then fixed to be the corresponding eigenvalue. (As noted above, the probability of ending up in a particular eigenstate is regulated by the coefficient of that eigenstate in the original wavefunction decomposition.)

Expectation value

A given operator, e.g. , implies an expectation value of:

     

See Also