Question

Пожалуйста решите уравнение Шрёдингера . дам 60 баллов честно за самый логичный ответ.

Answer 1

Где \hbar — постоянная Планка, \! m — масса частицы, \! U(x) — потенциальная энергия, \! E — полная энергия, \! \psi(x) — волновая функция. Для полной постановки задачи о нахождении решения \! ( 1 ) надо задать также граничные условия, которые представляются в общем виде для интервала \! [a,b]

\alpha_1\psi(a)+\beta_1\frac{d\psi(a)}{dx}=\gamma_1, \qquad ( 2 )
\alpha_2\psi(b)+\beta_2\frac{d\psi(b)}{dx}=\gamma_2, \qquad ( 3 )
где \! \alpha_1, \alpha_2, \beta_1, \beta_2, \gamma_1, \gamma_2 — константы. Квантовая механика рассматривает решения уравнения \! ( 1 ), с граничными условиями \! ( 2 ) и \! ( 3 ).

Answer 2

Уравнение Шрёдингера является основным уравнением квантовой механики, описывающим эволюцию квантовой системы во времени. Оно было предложено Эрвином Шрёдингером в 1925 году и является одним из ключевых достижений в физике.

Solving the Schrödinger Equation

To solve the Schrödinger equation, we need to consider the context in which it is being applied. The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is used to find the wave function of a quantum system and predict its behavior.

The general form of the time-dependent Schrödinger equation is given by:

iħ ∂Ψ/∂t = HΨ

Where: - i is the imaginary unit, - ħ is the reduced Planck constant, - Ψ is the wave function, - t is time, and - H is the Hamiltonian operator.

The Hamiltonian operator represents the total energy of the system.

Applying the Schrödinger Equation

To apply the Schrödinger equation to a specific physical system, we need to define the Hamiltonian operator H for that system. The form of the Hamiltonian operator depends on the system under consideration, such as a particle in a potential well, a harmonic oscillator, or a hydrogen atom.

The Hamiltonian operator for a particle of mass m in a potential V(x) in one dimension is given by:

H = - (ħ^2 / 2m) ∇^2 + V(x)

Where: - ∇^2 is the Laplacian operator, - m is the mass of the particle, and - V(x) is the potential energy function.

Conclusion

In conclusion, to solve the Schrödinger equation for a specific physical system, we need to define the Hamiltonian operator H for that system and then apply the time-dependent Schrödinger equation to find the wave function Ψ. The specific form of the Hamiltonian operator depends on the system under consideration, and it incorporates the kinetic and potential energy terms relevant to that system.

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