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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 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 $(q_1, q_2, \ldots, q_f, p_1, p_2, \ldos, p_f)$.

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:(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
$E$
E
Energía del sistema
J
$C$
C
Factor de normalización
-
$\Delta p$
Dp
Incerteza en el momento
kg m/s
$\Delta q$
Dq
Incerteza en la posición
m
$m$
m
Masa de la partícula
kg
$\vec{p}_i$
&p_i
Momento de la i-esima partícula
J
$\Omega(E,N)$
Omega_EN
Numero de estados con energía y partículas
-
$\Omega(E,N)$
Omega_EN
Numero de estados para energía y partículas dadas
-
$N$
N
Numero de Partículas
-
$h$
h
Planck constant
Js
$\vec{q}$
&q
Posición
m
$V$
V
Volumen
m^3

Calculations

First, select the equation:   to ,  then, select the variable:   to 
Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used


Equations

Examples

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)

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)

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 $(q_1, q_2, \ldots, q_f, p_1, p_2, \ldos, p_f)$.

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 10580)

ID:(435, 0)