Druyd:Knowledge

Example of free particles

Storyboard

Once we have defined the method for counting states and estimating probabilities in situations of interest, we can delve into how a system of many free particles behaves.

>Model

ID:(435, 0)

Case Classical Mechanics

Definition

In classical mechanics, a system is described by the coordinates $q_1, q_2, \ldots, q_f$ and momenta $p_1, p_2, \ldots, p_f$ , where $f$ represents the number of degrees of freedom. The state of the system is represented as a point in phase space, given by $(q_1, q_2, \ldos, q_f, p_1, p_2, \ldos, p_f)$ .

In the case of a system consisting of $N$ free particles, described using a total of $3N$ coordinates, the number of degrees of freedom is defined as $f = 3N$ .

ID:(524, 0)

Case Quantum Mechanics

Image

In quantum mechanics, the state is described by the wave function $\psi$ , which depends on the variables $q_1, q_2, \ldos, q_f$ , where $f$ is the number of degrees of freedom of the system.

The wave function is a solution, in the non-relativistic case and for particles without spin, of the Schrödinger equation. Eigenvalues are associated with wave functions, which typically depend on integers. These integers represent possible states of the system that are bounded by the system's energy.

ID:(523, 0)

Calculation of the number of states

Note

In the case of a system consisting of $N$ free particles, described using a total of $3N$ coordinates, the number of degrees of freedom is defined as $f = 3N$ .

ID:(10580, 0)

Example of free particles

Description

Once we have defined the method for counting states and estimating probabilities in situations of interest, we can delve into how a system of many free particles behaves.

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$E$

Energía del sistema

$C$

Factor de normalización

$\Delta p$

Incerteza en el momento

kg m/s

$\Delta q$

Incerteza en la posición

$m$

Masa de la partícula

$\vec{p}_i$

&p_i

Momento de la i-esima partícula

$\Omega(E,N)$

Omega_EN

Numero de estados con energía y partículas

$\Omega(E,N)$

Omega_EN

Numero de estados para energía y partículas dadas

$N$

Numero de Partículas

$h$

Planck constant

$\vec{q}$

Posición

$V$

Volumen

m^3

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

$V=\displaystyle\int_V d^3q$

V=\displaystyle\int_V d^3q

(ID 522)

$\Delta p\Delta q \sim h$

Dp * Dq = h

(ID 527)

$2mE=\displaystyle\sum_i^N\vec{p}_i^2$

2* m * E =@SUM( &p_i ^2, i ,1, N )

(ID 528)

$\Omega(E,N)=\Omega_0\left(\displaystyle\frac{V}{\Delta q^3}\right)^N\left(\displaystyle\frac{2mE}{\Delta p^2}\right)^{3N/2}$

\Omega(E,N)=\Omega_0\left(B\displaystyle\frac{V}{\Delta q^3}\right)^N\left(\displaystyle\frac{2mE}{\Delta p^2}\right)^{3N/2}

(ID 3433)

$\Omega = \Omega_0 \left(\displaystyle\frac{2 m }{ h ^2}\right)^{3 N /2} V ^ N E ^{3N/2}$

Omega = Omega_0 *((2* m / h ^2))^(3* N /2)* V ^ N * E ^(3* N /2)

(ID 4805)

Examples

Quantum Calculation of Number of States

Para poder calcular probabilidades debemos contabilizar las veces que una situaci n se da. En este caso podemos discretizar el espacio de fase en intervalos de largo \Delta q en la posici n y \Delta p en el momento.

Cada una de estas celdas es de un 'volumen' \Delta p\Delta q que se puede asumir debe ser del orden de la constante de Planck que es un an logo del principio de Heisenberg.

Por ello con se tiene que

$\Delta p\Delta q \sim h$

Surge asi una red de puntos discretos en el espacio de fase. Cada sistema esta en un estado que corresponde a uno de estos puntos.

La probabilidad de que el sistema se encuentre en un punto en particular se determina contabilizando los sistemas del ensamble que cumplen esta condici n dividido por todos los posibles estados.

(ID 527)

Case Classical Mechanics

In the case of a system consisting of $N$ free particles, described using a total of $3N$ coordinates, the number of degrees of freedom is defined as $f = 3N$ .

(ID 524)

Case Quantum Mechanics

In quantum mechanics, the state is described by the wave function $\psi$ , which depends on the variables $q_1, q_2, \ldos, q_f$ , where $f$ is the number of degrees of freedom of the system.

The wave function is a solution, in the non-relativistic case and for particles without spin, of the Schr dinger equation. Eigenvalues are associated with wave functions, which typically depend on integers. These integers represent possible states of the system that are bounded by the system's energy.

(ID 523)

Case of a free particle gas, volume

In the case of free particles, there is no positional dependence, and when calculating the phase space, it is necessary to sum or integrate over all positions.

Therefore, using the notation of , we have:

$V=\displaystyle\int_V d^3q$

(ID 522)

Calculation of the number of states

In the case of a system consisting of $N$ free particles, described using a total of $3N$ coordinates, the number of degrees of freedom is defined as $f = 3N$ .

(ID 10580)

Case of a free particle gas

In the case of $N$ free particles in the classical approximation, we need to perform an integration in phase space with the constraint that describes the system.

Since it's a gas of free particles, the constraint is solely related to energy and doesn't depend on position. Therefore, we can express the energy as:

$E=\displaystyle\frac{1}{2m}\displaystyle\sum_i^N\vec{p}_i^2$

This expression can be simplified using mathematical notation as:

$2mE=\displaystyle\sum_i^N\vec{p}_i^2$

This formula represents a "sphere of radius" $\sqrt{2mE}$ in a "phase space" of $3N$ dimensions.

(ID 528)

Number of states of a gas of free particle

La integral sobre el espacio de estados se calcula con posición $m$ and volumen $m^3$ mediante

$V=\displaystyle\int_V d^3q$

con la condici n para la energ a que con energía del sistema $J$ , masa de la partícula $kg$ , momento de la i-esima partícula $J$ and numero de Partículas $-$ es

$2mE=\displaystyle\sum_i^N\vec{p}_i^2$

\\n\\nse deja integrar en forma simple en las coordenadas espaciales. Cada integral es igual al volumen V y como son N part culas dichas integrales dan V^N.\\n\\nLa integral sobre el momento se limita a una la superficie de una esfera de radio \sqrt{2mE}. En analog a al caso tridimensional la superficie sera proporcional al radio elevado al numero de grados de libertad 3N menos uno, lo que igual se puede asumir como:\\n\\n

$3N-1\sim 3N$

Por ello se puede estimar con energía del sistema $J$ , masa de la partícula $kg$ , momento de la i-esima partícula $J$ and numero de Partículas $-$

$\Omega(E,N)=\Omega_0\left(\displaystyle\frac{V}{\Delta q^3}\right)^N\left(\displaystyle\frac{2mE}{\Delta p^2}\right)^{3N/2}$

(ID 3433)

Number of free particle states with cell

En el calculo del n mero de estados se obtiene el n mero de estados con energía del sistema $J$ , factor de normalización $-$ , incerteza en el momento $kg m/s$ , incerteza en la posición $m$ , masa de la partícula $kg$ , numero de estados para energía y partículas dadas $-$ , numero de Partículas $-$ and volumen $m^3$ son

$\Omega(E,N)=\Omega_0\left(\displaystyle\frac{V}{\Delta q^3}\right)^N\left(\displaystyle\frac{2mE}{\Delta p^2}\right)^{3N/2}$

Como el elemento de volumen del espacio de fase es con incerteza en el momento $kg m/s$ , incerteza en la posición $m$ and planck constant $J s$ igual a

$\Delta p\Delta q \sim h$

por lo que el n mero de estados se deja simplificar con incerteza en el momento $kg m/s$ , incerteza en la posición $m$ and planck constant $J s$ a

$\Omega = \Omega_0 \left(\displaystyle\frac{2 m }{ h ^2}\right)^{3 N /2} V ^ N E ^{3N/2}$

(ID 4805)

ID:(435, 0)