Derivation of Schrodinger Equation for Hydrogen Atom
The derivation of the Schrödinger equation for the hydrogen atom
The principle of all things is subtle and unpredictable. In the domain of atoms, the wonder of quantum is especially evident. To deduce the Schrödinger equation for the hydrogen atom today, it is necessary to understand the basis of its physics and follow the path of mathematics.
The hydrogen atom is composed of a proton and an electron. Electrons move around the proton, and the interaction between them is dominated by the Coulomb force. In the context of classical mechanics, the movement of electrons follows the rules of orbit, but the quantum world is not complete.
In quantum mechanics, the state of microscopic particles is described by the wave function. The modulus of the wave function | 🥰 | ² represents the probability density of particles appearing at a certain point in time and space.
First discuss the energy of the system. The energy E of the hydrogen atomic system consists of the electron kinetic energy\ (T\) and the electric potential energy\ (V\). The mass of the electron is\ (m\), and its kinetic energy is\ (T =\ frac {p ^ {2}} {2m}\), where\ (p\) is the electron momentum. The electric potential energy\ (V\) is generated by the Coulomb interaction between protons and electrons. The Coulomb constant is\ (k\), and the charges of electrons and protons are\ (-e\) and\ (e\), respectively. If the distance between the two is\ (r\), then\ (V = -\ frac {k e ^ {2}} {r}\). Therefore, the system energy\ (E = T + V =\ frac {p ^ {2}} {2m} -\ frac {k e ^ {2}} {r}\).
According to de Broglie's wave-particle duality, particles have the property of waves, and the momentum\ (p\) is related to the wavelength\ (\ lambda\), that is,\ (p =\ frac {h} {\ lambda}\), where\ (h\) is Planck's constant.
In quantum mechanics, observable measurements are expressed as operators. The momentum operator\ (\ hat {p} = -i\ hbar\ nabla\), where\ (\ hbar =\ frac {h} {2\ p i }\),\(\ nabla\) is the layer operator.
Substitute the momentum operator into the energy expression to obtain the Hamiltonian operator\ (\ hat {H} = -\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2} -\ frac {k e ^ {2}} {r}\), where\ (\ nabla ^ {2}\) is the Laplace operator.
The fundamental equation of quantum mechanics, the Schrodinger equation, has the form\ (\ hat {H}\ Psi = E\ Psi\).
For the hydrogen atom, the Hamiltonian operator is substituted into the Schrodinger equation to obtain the Schrodinger equation of the hydrogen atom:\ (-\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2}\ Psi -\ frac {k e ^ {2}} {r}\ Psi = E\ Psi\).
This equation is in the spherical coordinate system, and can be separated by variables due to the spherical symmetry of the hydrogen atom. Let\ (\ Psi (r,\ theta,\ varphi) = R (r) Y (\ theta,\ varphi) \) be substituted into the equation, and after some mathematical derivation, the radial equation and the angular equation can be obtained.
The radial equation is related to the change of the distance\ (r\) between the electron and the proton, while the angular equation determines the dependence of the wave function on the angles\ (\ theta\) and\ (\ varphi\).
Solving the Schrodinger equation of the hydrogen atom can obtain the possible energy states and wave function distributions of the electron in the hydrogen atom. This is the basis for understanding the structure and properties of the hydrogen atom, and is also an important application of quantum mechanics in atomic physics.
The principle of all things is subtle and unpredictable. In the domain of atoms, the wonder of quantum is especially evident. To deduce the Schrödinger equation for the hydrogen atom today, it is necessary to understand the basis of its physics and follow the path of mathematics.
The hydrogen atom is composed of a proton and an electron. Electrons move around the proton, and the interaction between them is dominated by the Coulomb force. In the context of classical mechanics, the movement of electrons follows the rules of orbit, but the quantum world is not complete.
In quantum mechanics, the state of microscopic particles is described by the wave function. The modulus of the wave function | 🥰 | ² represents the probability density of particles appearing at a certain point in time and space.
First discuss the energy of the system. The energy E of the hydrogen atomic system consists of the electron kinetic energy\ (T\) and the electric potential energy\ (V\). The mass of the electron is\ (m\), and its kinetic energy is\ (T =\ frac {p ^ {2}} {2m}\), where\ (p\) is the electron momentum. The electric potential energy\ (V\) is generated by the Coulomb interaction between protons and electrons. The Coulomb constant is\ (k\), and the charges of electrons and protons are\ (-e\) and\ (e\), respectively. If the distance between the two is\ (r\), then\ (V = -\ frac {k e ^ {2}} {r}\). Therefore, the system energy\ (E = T + V =\ frac {p ^ {2}} {2m} -\ frac {k e ^ {2}} {r}\).
According to de Broglie's wave-particle duality, particles have the property of waves, and the momentum\ (p\) is related to the wavelength\ (\ lambda\), that is,\ (p =\ frac {h} {\ lambda}\), where\ (h\) is Planck's constant.
In quantum mechanics, observable measurements are expressed as operators. The momentum operator\ (\ hat {p} = -i\ hbar\ nabla\), where\ (\ hbar =\ frac {h} {2\ p i }\),\(\ nabla\) is the layer operator.
Substitute the momentum operator into the energy expression to obtain the Hamiltonian operator\ (\ hat {H} = -\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2} -\ frac {k e ^ {2}} {r}\), where\ (\ nabla ^ {2}\) is the Laplace operator.
The fundamental equation of quantum mechanics, the Schrodinger equation, has the form\ (\ hat {H}\ Psi = E\ Psi\).
For the hydrogen atom, the Hamiltonian operator is substituted into the Schrodinger equation to obtain the Schrodinger equation of the hydrogen atom:\ (-\ frac {\ hbar ^ {2}} {2m}\ nabla ^ {2}\ Psi -\ frac {k e ^ {2}} {r}\ Psi = E\ Psi\).
This equation is in the spherical coordinate system, and can be separated by variables due to the spherical symmetry of the hydrogen atom. Let\ (\ Psi (r,\ theta,\ varphi) = R (r) Y (\ theta,\ varphi) \) be substituted into the equation, and after some mathematical derivation, the radial equation and the angular equation can be obtained.
The radial equation is related to the change of the distance\ (r\) between the electron and the proton, while the angular equation determines the dependence of the wave function on the angles\ (\ theta\) and\ (\ varphi\).
Solving the Schrodinger equation of the hydrogen atom can obtain the possible energy states and wave function distributions of the electron in the hydrogen atom. This is the basis for understanding the structure and properties of the hydrogen atom, and is also an important application of quantum mechanics in atomic physics.

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