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Schrodinger Equation For Hydrogen Atom
On the Schrödinger Equation of Hydrogen Atoms
Between heaven and earth, the principle of all things is tiny and mysterious. In the microscopic domain, the wonders of hydrogen atoms are explored by the Schrödinger equation.
Hydrogen atoms are the basis of atoms. Its Schrödinger equation is the standard for describing the motion state of electrons in hydrogen atoms. This equation is based on the wave function $\ Psi $, which is related to the probability distribution of electrons.
The form of the equation, including time, is $i\ hbar\ frac {\ partial\ Psi} {\ partial t} = -\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2}\ Psi + V\ Psi $, where $\ hbar $is the reduced Planck constant, $m $is the electron mass, and $V $is the electron potential energy. For a hydrogen atom, $V = -\ frac {e ^ {2}} {4\ pi\ epsilon_ {0} r} $, $e $is the electron charge, $\ epsilon_ {0} $is the vacuum dielectric constant, and $r $is the distance between the electron and the nucleus.
The Schrodinger equation without time is more commonly used, that is, $-\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2}\ psi + V\ psi = E\ psi $. Solving this equation, the energy level and wave function of the hydrogen atom can be obtained. The energy level $E_ {n} = -\ frac {13.6} {n ^ {2}} eV $, $n = 1,2,3,\ cdots $, which is the quantized energy level, which is not continuous.
The wave function $\ psi_ {nlm} $is determined by the principal quantum number $n $, the angular quantum number $l $, and the magnetic quantum number $m $. For example, when n = 1 $, l = 0 $, m = 0 $, the wave function $\ psi_ {100} $describes the electronic state of a hydrogen atom in the ground state.
From the Schrödinger equation, the shape of the electron cloud of a hydrogen atom can be known, and the rules of the electron transition can also be known. Electrons transition from high energy levels to low energy levels, emitting light quanta; conversely, they absorb light quanta and transition.
Schrödinger equation in the study of hydrogen atoms has enabled our generation to glimpse the secrets of the microscopic world, making unparalleled contributions to the development of quantum physics and paving the way for future generations to explore the microscopic principles of matter.
Between heaven and earth, the principle of all things is tiny and mysterious. In the microscopic domain, the wonders of hydrogen atoms are explored by the Schrödinger equation.
Hydrogen atoms are the basis of atoms. Its Schrödinger equation is the standard for describing the motion state of electrons in hydrogen atoms. This equation is based on the wave function $\ Psi $, which is related to the probability distribution of electrons.
The form of the equation, including time, is $i\ hbar\ frac {\ partial\ Psi} {\ partial t} = -\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2}\ Psi + V\ Psi $, where $\ hbar $is the reduced Planck constant, $m $is the electron mass, and $V $is the electron potential energy. For a hydrogen atom, $V = -\ frac {e ^ {2}} {4\ pi\ epsilon_ {0} r} $, $e $is the electron charge, $\ epsilon_ {0} $is the vacuum dielectric constant, and $r $is the distance between the electron and the nucleus.
The Schrodinger equation without time is more commonly used, that is, $-\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2}\ psi + V\ psi = E\ psi $. Solving this equation, the energy level and wave function of the hydrogen atom can be obtained. The energy level $E_ {n} = -\ frac {13.6} {n ^ {2}} eV $, $n = 1,2,3,\ cdots $, which is the quantized energy level, which is not continuous.
The wave function $\ psi_ {nlm} $is determined by the principal quantum number $n $, the angular quantum number $l $, and the magnetic quantum number $m $. For example, when n = 1 $, l = 0 $, m = 0 $, the wave function $\ psi_ {100} $describes the electronic state of a hydrogen atom in the ground state.
From the Schrödinger equation, the shape of the electron cloud of a hydrogen atom can be known, and the rules of the electron transition can also be known. Electrons transition from high energy levels to low energy levels, emitting light quanta; conversely, they absorb light quanta and transition.
Schrödinger equation in the study of hydrogen atoms has enabled our generation to glimpse the secrets of the microscopic world, making unparalleled contributions to the development of quantum physics and paving the way for future generations to explore the microscopic principles of matter.
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