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Консультация № 159934
08.02.2009, 14:55
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Здравствуйте эксперты, помогите пожалуйста с задачкой:
Частица находиться в бесконечной однородной потенциальной яме шириной L в первом возбужденном состоянии. В каких точках ямы плотность вероятности обнаружения частицы максимальна, а в каких минимальна?
Частица находиться в бесконечной однородной потенциальной яме шириной L в первом возбужденном состоянии. В каких точках ямы плотность вероятности обнаружения частицы максимальна, а в каких минимальна?
Обсуждение
Неизвестный
09.02.2009, 09:41
общий
это ответ
Здравствуйте, Tribak!
Решение уравнения Шредингера для частицы в одномерной прямоугольной бесконечно глубокой потенциальной яме шириной L дает собственные функции psi = sqrt(2/L) * sin(n*pi*x/L), где x меняется от 0 до L. Плотность вероятности (обозначим ее F) является квадратом собственной функции. F = 2/L* sin^2(n*pi*x/L), здесь sin^2 - квадратный синус.
Задача нахождения точек максимумов и минимумов решается стандартным образом - экстремумы находятся в точках, где первая производная (F') равна нулю: F' = sin(2*n*pi*x/L)=0, откуда находим точки экстремумов xm = m*L / (2*n), где m - целое число, а xm находятся в пределах ширины ящика. Первое возбужденное состояние соответствует n=2, тогда точки экстремумов xm = 0; L/4; L/2; 3L/4; L.
Определить максимумы и минимумы можно по знаку второй производной (F'') в точке экстремума. F''<0 для максимума и F''>0 для минимума. F''=cos(2*n*pi*x/L) = cos(4*pi*x/L). Подставляя сюда координаты xm точек экстремумов, найдем, что xm = 0; L/2; L - минимумы, а xm = L/4; 3L/4 - максимумы.
Решение уравнения Шредингера для частицы в одномерной прямоугольной бесконечно глубокой потенциальной яме шириной L дает собственные функции psi = sqrt(2/L) * sin(n*pi*x/L), где x меняется от 0 до L. Плотность вероятности (обозначим ее F) является квадратом собственной функции. F = 2/L* sin^2(n*pi*x/L), здесь sin^2 - квадратный синус.
Задача нахождения точек максимумов и минимумов решается стандартным образом - экстремумы находятся в точках, где первая производная (F') равна нулю: F' = sin(2*n*pi*x/L)=0, откуда находим точки экстремумов xm = m*L / (2*n), где m - целое число, а xm находятся в пределах ширины ящика. Первое возбужденное состояние соответствует n=2, тогда точки экстремумов xm = 0; L/4; L/2; 3L/4; L.
Определить максимумы и минимумы можно по знаку второй производной (F'') в точке экстремума. F''<0 для максимума и F''>0 для минимума. F''=cos(2*n*pi*x/L) = cos(4*pi*x/L). Подставляя сюда координаты xm точек экстремумов, найдем, что xm = 0; L/2; L - минимумы, а xm = L/4; 3L/4 - максимумы.
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