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Hamiltonian function with summation and potential energy terms

Answer:
The Hamiltonian \( \hat{H} \) is expressed as the sum of kinetic and potential energy terms, specifically the sum over \( N \) particles of their kinetic energy operators plus a potential energy function \( V \). It is given by:

\[ \hat{H} = - \frac{\hbar^2}{2m} \sum_{n=1}^N \frac{\partial^2}{\partial x_n^2} + V(x_1, x_2, \ldots, x_N, t) \]

Explanation:
This expression represents the Hamiltonian operator in quantum mechanics for a system of \( N \) particles in one dimension. The first term corresponds to the total kinetic energy, derived from the classical kinetic energy but expressed as a differential operator (the Laplacian in one dimension). The second term is the potential energy, which can depend on the positions of all particles and possibly time. The key concepts involved are the Schrödinger Hamiltonian, the kinetic energy operator, and potential energy functions.

The kinetic energy operator for each particle is derived from the classical form \( \frac{p^2}{2m} \), where \( p \) is the momentum operator \( -i \hbar \frac{\partial}{\partial x} \). The total Hamiltonian sums these over all particles, leading to the sum of second derivatives with respect to each position coordinate.

Steps:

  1. Identify the kinetic energy operator for a single particle:

In quantum mechanics, the momentum operator in one dimension is:

\[ \hat{p}_n = -i \hbar \frac{\partial}{\partial x_n} \]

The kinetic energy operator for the \( n \)-th particle is:

\[ \hat{T}_n = \frac{\hat{p}_n^2}{2m} = - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x_n^2} \]

  1. Sum over all particles:

The total kinetic energy operator for the system is:

\[ \hat{T} = \sum_{n=1}^N \hat{T}_n = - \frac{\hbar^2}{2m} \sum_{n=1}^N \frac{\partial^2}{\partial x_n^2} \]

  1. Add the potential energy term:

The potential energy \( V \) can depend on all particle positions and time:

\[ V(x_1, x_2, \ldots, x_N, t) \]

  1. Construct the Hamiltonian:

Combining kinetic and potential energy operators, the Hamiltonian operator becomes:

\[ \hat{H} = \hat{T} + V = - \frac{\hbar^2}{2m} \sum_{n=1}^N \frac{\partial^2}{\partial x_n^2} + V(x_1, x_2, \ldots, x_N, t) \]

  1. Express the Hamiltonian explicitly:

The full expression matches the given formula, confirming the derivation.

Summary:
The formula is the standard form of the many-particle Hamiltonian in non-relativistic quantum mechanics, involving the kinetic energy operators (via the Laplacian/double derivative) and the potential energy function. The key theorems involved are the representation of momentum and kinetic energy operators in quantum mechanics, and the principle that the Hamiltonian governs the evolution of the quantum state.

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