Quantum Mechanics
Quantum mechanics is a theory that describes the interactions of all particles and systems. It underlies all physical phenomena, including scattering.
Contents
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 positionspace or momentumspace:
These representations can be interrelated (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.
A basis should be orthonormal:
Integral Notation  Dirac Notation  

normalized  
orthogonal 
An operator defines a particularly 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 (braket 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
A wave packet is a localized wavelike perturbation. Particles in quantum mechanics can be thought of as wavepackets.
Note that "waveparticle duality" can be misleading. One can imagine a quantum particle as "both a wave and a particle"; however, it might be better to instead imagine it as a "wave packet". The 'particle' and 'wave' descriptions are really idealized limiting cases, which never appear in reality:
 A classical 'particle' is a pointlike object. In QM would have a corresponding infinite spread in its momentum. Such an idealized (infinitely small) entity cannot truly exist.
 A classical 'wave' is a plane wave: an oscillation with a perfectly welldefined wavelength, extending infinitely in both directions. In QM, we indeed note that having a preciselydefined wavelength (momentum) implies infinite spatial spread (i.e. the wave fills the entire universe). Such a construct is not physicallyrealizable.
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:
where and are complexvalued coefficients. 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  Matrix Notation 

Operators
Observables in QM appears as operators ().
Examples:
Operator  Position basis  Momentum basis 

Position  
Momentum  
Hamiltonian  
Energy 
Commutators
The commutator is a useful function for evaluating quantum objects. It is defined by:
The commutator can be thought of as a composite operator, which can thus act on a wavefunction:
The uncertainty relations are intimately tied to commutators of operators. For instance, for two noncommuting operators:
Then the related uncertainties follow:
Therefore, noncommuting operators have a indeterminacy relation between their associated observables; and the commutation relation provides the bound on this interrelation. The two observables cannot both be simultaneously definite (the wavefunction cannot be in a state that is simultaneously an eigenstate of both operators, so it cannot have a definite eigenvalue for both of the observables). On the other hand, if two operators commute:
Then the wavefunction can be in a state that is simultaneously an eigenstate of both operators. There is thus no restriction on both observables (A and B) being simultaneously definite (known to arbitrary precision).
Measurement
For state and linear map :
Integral Notation  Dirac Notation  Matrix Notation 



The linear map is a function (in the complex vector space , dual to ) that maps from vector space to . In conventional notation this operation would be performed using a function applied to :
In a quantum measurement, the transition of the wavefunction into state can be thought of as:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \phi  \psi \rangle = a_1^*c_1 + a_2^*c_2 + a_3^*c_3 + \cdots } 
Thus is the probability amplitude for the initial state to 'collapse' into the state . Measurements take the form of operators in QM. 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 (for state ) of:
If the system is in an eigenstate of the operator:
We know that:
And so:
In other words, the expectation value of an eigenstate is simply the eigenvalue.
Schrödinger Equation
Timeindependent equation
This simplified version of the Schrödinger equation can be used to solve for allowed stationary states. The general form is akin to the eigenvalue problems noted above: the energy operator () acts on the system state () to yield an energy eigenvalue ():
For a single nonrelativistic particle, the Hamiltonian is known and the Schrödinger equation takes the form:
Timedependent equation
More generally, the timeevolution of the wavefunction should be considered. The full version of the Schrödinger equation thus includes time dependence:
Again for a single nonrelativistic particle, we can write more specifically that:
Entanglement
When systems (or Hilbert spaces) A and B interact, they become entangled. Before the interaction, the two systems are simply a composite system:
At this level, states are separable. However, the composite system more generally should be written as:
If all the coefficients can be written as , then there is no coupling between the two states. We call the system separable, since it can be decomposed into the two subsystems:
However, if , then the states are nonseparable, or entangled.
The entanglement of an observer (or simply measurement apparatus) can also be considered. For observer performing measurement on system :
This leads to the evolution:
Where the righthandside describes an entanglement between the observer and the system being studied.
Density Matrices
The outer product of a ket with a bra defines a 2D matrix; i.e. a linear operator.
Density matrices can be a useful way to visualize the interactions between states of a system, and timeevolution. Thus we can define the density matrix as:
The elements of the density matrix can be interpreted as a form of quantum probability distribution. For instance, a pure state is given by a density matrix that has only a single nonzero term, along the diagonal:
By comparison, a mixed state is one where more than one diagonal term is nonzero, but offdiagonal terms are zero:
The diagonal terms of the density matrix will be real numbers (and can be interpreted as probabilities). The above matrix describes a classical mixture of different states. In the Copenhagen interpretation, this would be described as a situation where the system will randomly collapse into one of the possible states. Under MWI, it would instead be said that the various states are now noninteracting, and thus will evolve independently.
The most general case is where the offdiagonal terms are nonzero. These terms represent the interference aspects. Thus this describes an entangled state:
The offdiagonal terms indicate the strength and nature of the interference between the states. Thus, the diagonal elements () capture the probability of occupying a certain state as The timedependence often appears in these offdiagonal () terms as .
During decoherence, these offdiagonal terms are driven towards zero.
Decoherence
TBD