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Dmytro Taranovsky
Created: August 2, 2000
Last Modified:January 18, 2002
Minor Editing: áJanuary 28, 2003

Quantum Mechanics

In quantum mechanics, each system is in a superposition of its basis states. For example, particles assume a superposition of all positions r and (using a different basis) a superposition of momenta p. The principle of superposition applies, so time evolution is linear. A vector is a superposition of the basis vectors, so quantum systems are represented by vectors. The quantum state of the universe can be approximated to the desired precision by a vector in a finite-dimensional vector space and can be stated exactly either as a sequence of such vectors by using increasingly detailed approximations or as a vector in an infinite dimensional Euclidean space.
Note: h-bar has been omitted from the equations in accordance with Foundations of Physics.

Each system is represented by a vector (with complex-valued components) normalized to one. Observation is represented by a Hermitian (Oab=Oba*) matrix/operator O. As a result of observation, the vector becomes one of the orthonormal eigenvectors of O. ("a" is normalized to 1 means <a, a>=1. A set of vectors is called orthonormal when every vector is normalized to 1 and for every 2 different vectors their inner product is 0.) The observation gives an eigenvalue (λ) corresponding to the eigenvector. The amplitude of the transfer from p to q, ψ =<q|p> = <p, q> = sum of pjqjc for all j such that pjand qj are components of p and q. (for real a and b, (a+bi)c= (a-bi); i2=-1). (For functions, <p, q>= integral of p(r)q(r)c over the domain, where r is the argument of the functions.) The probability of the transfer to a given eigenvector, P, is |ψ|2. The expectation value is S(Piλi) where the summation is over all eigenvectors and can be shown to equal <p|Op> for state p. According to the principle of superposition, observation should yield a superposition of results, not a single result. Thus, quantum mechanics cannot apply completely to the observer. (Quantum mechanics can, however, be applied to an "observer" that cannot feel by stating that an observation leads to an entanglement (relation) between the observer and the system observed.)

Time dependence of systems is described by the linear differential equation: i*φ′(t)=Hφ where φ(t) is the state at time t and H, called Hamiltonian, is a Hermitian matrix/operator. (i2=-1). Therefore, φ(t)=e-iHtφ(0). Hamiltonian is an observable--it is energy.

Theorem: Quantity represented by O is conserved if and only if physical laws are invariant under the operator eiλO for every real λ.
Corollary: Setting O=H, invariance of physical laws in time (since e-iHt is time translation) shows conservation of energy.
Note: Conserved quantities have the same eigenvectors as Hamiltonian (H) and thus commute with H.
Specific Example: If a system is isolated, H does not depend on time since dφ/dt depends only on φ and physical laws, and physical laws are invariant of time. If H is invariant of time, and if a state (φi) is an eigenvector of H, the time dependence can be shown to be e-iEtφi where E is energy of the system, which is the corresponding eigenvalue. Since the probability distribution of φi does not change in time, it is called a stationary state. Since Hermitian matrices have a full set of orthonormal eigenvectors, every system is a superposition of stationary states with coefficients of time independent magnitude. Thus, (and since energy of stationary states is time independent) energy is conserved.

Sometimes, certain quantum states need not be distinguished. In that case, the basis states are arranged in several collections, such that states in a collection are not distinguished but the collections themselves are distinguished. The collections can then be considered as basis states. (The amplitude for a collection is the sum of the amplitudes for the basis states of the collection.) An observation can be partial, that is treating collections as basis states. A partial observation does not affect the relative value of states in a collection.

Quantum mechanics provides the framework for individual theories. A quantum theory is created by stating information about the Hamiltonian and relating the basis states to human observations.

We now build the quantum theory for non-relativistic particles. A finite collection of particles is given. Each particle has a type (such as electron or proton). Each type is either fermion or boson. Each state is a doubly differentiable superposition of the following states (r):

Each particle has a position (an ordered triple of coordinates) and a spin (either true or false). (Thus, a state for a system of n particles has 3n real valued components and n Boolean components.) We are given the total potential energy of the system (U) as a function of the state (r).

The quantum state is subject to the following restriction: An interchange of any two fermion particles of the same type multiples the state by -1. An interchange of boson particles of the same type does not change the state. (According to quantum mechanics, particles of the same type are identical and their interchange cannot be observed.)

The Hamiltonian H is the sum of the kinetic and potential part. The potential part is the operator U, that is (Uφ)(r)=U(r)φ(r). Let φ(i, j) be the distribution of coordinate j (one of the 3 spatial coordinates) of particle i. It is computed by φ(i, j)(x)=the appropriate integral of φ(r) over the region where the coordinate j of particle i is equal to x. (Theorem:áφ(i, j) is normalized.)á The kinetic part is the sum of K(i, j) (kinetic energy operator) for all particles i and coordinates j, where K(i, j)(r)=-(∂2φ(i, j)(x) /∂x2)/2mi where x is the coordinate j of particle i and mi is mass of particle i.áThe basic theory is now stated and can be specified further.

Thus, for a single non-relativistic particle of mass m, H=p2/2m+U, where U(r) is potential energy at r, and momentum, p=-i(∂/∂x+∂/∂y+∂/∂z). (For a free non-relativistic particle with energy E and momentum p, the wavefunction is φ(r,t) = C*e-i(Et-pĚr).)

Two fermions of the same type cannot assume the same quantum state (the superposition of the 3 coordinates and the spin) because then their exchange would not affect the system and thus will not multiply the state by -1. It can be proved that because spin is a superposition of two values, at most two fermions of the same type can assume the same superposition of coordinates. Other fermions may have to assume states of higher energy.

Energy is quickly observed. Thus, systems tend to assume states of definitive energy. In reality (but not in this theory), systems can emit and absorb photons (light) and change between states of different energy. The energy of the photon can be found from the conservation of energy. Spontaneously, systems eventually emit photon(s), if such emission is possible. The emission and absorption cannot be properly described here because photons move with a relativistic speed c. However, non-relativistic approximations for the transition rates exist.

The theory above is further specified by stating the particle types and the potential energy. The following specification correctly predicts large parts of chemistry. Each particle has a mass (mi,mi>0) and electric charge (a real number qi). The potential energy is the sum of potential energies (U{i, j}) for all unordered pairs {i, j} of particles ({i, i} is not counted). U{i, j}=kqiqj/d(i, j)-Gmimj/d(i, j) where d is distance between its arguments, k is the Coulomb's constant and G is the gravitational constant. (The second term is very small and can usually be ignored.) The particles are electrons and nuclei of atoms. Electron is a fermion and has electric charge of -1. Nuclei are classified by proton number (p) and neutron number (n).p and n are nonnegative integers; n+p>0. Charge of nucleus is p. Mass is roughly proportional to n+p; tables of nuclides have more precise masses; mass of a proton (p=1,n=0) is approximately 1.67*10-27kg; mass of an electron is about 9.11*10-31kg. A nucleus is a fermion if and only if n+p is odd.

Unfortunately, calculations are often difficult, so elaborate approximation techniques are used. For example, many systems are numerically analyzed as a small perturbation (deviation) from systems for which the time dependence was solved. Generalized functions are used as idealization of particular conditions (a sharp change in the potential is idealized as an instantaneous change). In non-relativistic mechanics, the stationary states are non-normalizable: The wavefunction density must be the same everywhere in space and hence cannot be normalized to one. Instead, the system is typically represented by the coordinates of the center of mass and the coordinates of all particles but one relative to the center of mass. The time evolution (dependence) of the system is then separated into time dependence of the center of mass (which can be easily computed since momentum is conserved) and the evolution of the coordinates relative to the center of mass of all particles but one. The later is usually normalizable, and the equations of motion can be derived from the theory.