Extrema
The variational Monte Carlo
Reason why VP is an approximation
Mean-field approximation
PDF supplement on Hamilton's variational principle
A maximization/minization problem is to find a value for each independent variable such that certain function/functional is maximized/minized. Calculus of variation utilize the idea that variations of a function/functional at its local extrema is zero. (Since the converse is not true, the solution is never complete for all maximization problems.) For examples:Constraints limits the degree of freedom (or the number of independent variables). Thus some variables can be eliminated from the function/functional before applying the calculus of variation.If x varies with another variable t, one could find all values of t such that the function dependent on x which therefore depends on t is maximized.
If x is itself a variable function dependent on t, there are many x(t) that can maximize the function F(x) for some t. There are some or one x(t) that could maximize the sum of F(x) over t.
If F is also dependent on t, there are some x(t) that could maximize F(x,t) for some t and there is also x(t) that could maximize the integral of F(x,t) over t.
Another perspective is to express the constrain as a function that needs to be stationary. Constrained maximization/minimization problem can be solved with Lagrange Multiplier.Maximization Problem with independent variables
Problem: Choose x and y independently such that F(x,y) is maximized.
solution: Partially differentiate F with respect to x and y and set them to zero.Constrainted Maximization Problem
Problem: Choose the pair of (x,y) that satisfy f(x,y)=c for some constant c such that F(x,y) is maximized.
solution 1 (Lagrangian approach): Partially differentiate F and f with respect to x and y and set them to equal within a scalar. Solve for (x,y) together with the constraint f(x,y)=c.
solution 2 (substitution approach): Eliminate x or y in F using the condition f(x,y)=c. Partially differentiate the new F with respect to the remaining independent variable and set it to zero.Minimization Problem by choosing a function
Problem: Choose x(t) such that integral with respect to t of F(x(t),t) is minimized. (It is called an action integral if F is the lagrangian of a physical system. The condiction that the action integral is minized is called the law of least action.)
solution: Partially differentiate F with respect to x and set it to zero. (same as applying lagrange equations). This could be interpreted as finding a path in the x-t plane such that the action integral is stationary.
In the last maximation problem, the action integral maps a function of t to a quantity that is independent of t. Likewise, a more general problem may involves other type of functions that eliminate t. The following illustrates that:Example: Orthogonality
Problem: Maximize the integral of x(t)cos(t) from -infinity to infinity with the constraint that x(t) must be a sinuisoid of amplitude a.Solution: By orthogonality, x(t)=acos(t) for any constant a.
Comment: This maximized the function up to a constant factor a. What is the solution without the constraint that x(t) being sinuisoid? The problem can also be viewed as choosing a frequency for x(t) such that the function is maximized. The degree of freedom reduces from infinite (x at each time) to 1.
Helium Ground State Energy
Calculus of variation is a useful technique used both in classical mechanics and quantum mechanics. In quantum mechanics, it is used to approximate the ground state energy of a physical system characterized by a wavefunction.
HamiltonianSchrödinger's equation
A partial differential equation that describes how the wavefunction of a physical system evolves over time.Hilbert Space
A vector that can be turned into a complete (a space of complete functions [a set of orthonormal functions such that the minimum sum of square of every piecewise continuous function approximated by the linear sum of orthonomal functions approaches zero as the total number of orthonormal functions used approaches infinite]) metric space with a proper definition of the inner product.Basis
Linearly independent vectors that span a vector space.Fermion
An odd half-integer spin particle which acts on each other by exchanging bosons (particle with symmetric wavefunctions [a scalar function used to describe the properties of a wave]). Examples include leptons (participate in weak, electromagnetic, and gravitational interactions, such as electron), neutrons, protons and quarks (which are the building blocks of hadrons (a particle which interacts via the strong force, consisting of the baryons [spin that is integer multiple of the h-bar/reduced Planck's constant] and mesons [even integer spin]) and interacts via strong, weak, electromagnetic or gravitational force).Functional
a quantity dependent on the behavior of one or many functions over an interval. An example is the action integral. It is a function of functions.
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Author: Chan, Chung Last modified: 23ndJuly,2002