The H^-ion consists of two electrons bound to a proton....

The $\mathrm{H}^{-}$ion consists of two electrons bound to a proton. Estimate its ground-state energy by adapting the calculation of helium's groundstate energy that uses the variational principle. Show that using singleparticle wavefunctions $u(\mathbf{x}) \propto \mathrm{e}^{-r / a}$ the expectation of the Hamiltonian is $$ \langle H\rangle_{a}=\mathcal{R}\left(2 x^{2}-\frac{11}{4} x\right) \quad \text { where } \quad x \equiv \frac{a_{0}}{a} $$ Hence find that the binding energy of $\mathrm{H}^{-}$is $\sim 0.945 \mathcal{R}$. Will $\mathrm{H}^{-}$be a stable ion?

The $\mathrm{H}^{-}$ion consists of two electrons bound to a proton. Estimate its ground-state energy by adapting the calculation of helium's groundstate energy that uses the variational principle. Show that using singleparticle wavefunctions $u(\mathbf{x}) \propto \mathrm{e}^{-r / a}$ the expectation of the Hamiltonian is
$$
\langle H\rangle_{a}=\mathcal{R}\left(2 x^{2}-\frac{11}{4} x\right) \quad \text { where } \quad x \equiv \frac{a_{0}}{a}
$$
Hence find that the binding energy of $\mathrm{H}^{-}$is $\sim 0.945 \mathcal{R}$. Will $\mathrm{H}^{-}$be a stable ion?

Key Concepts

Atomic Stability

Atomic stability refers to the ability of an atomic ion or molecule to remain bound under perturbations. In quantum systems, if the calculated binding energy is sufficiently negative, it indicates that the system is energetically stable. The discussion regarding the binding energy in such variational calculations helps determine whether an ion, like the system under study, can exist as a stable entity or is prone to dissociation.

Expectation Value of the Hamiltonian

The expectation value of the Hamiltonian is the average energy calculated over a given wavefunction. In the context of the variational method, it is computed by integrating the product of the trial wavefunction, the Hamiltonian operator, and the complex conjugate of the trial wavefunction over all space. This quantity is minimized with respect to the variational parameters to obtain the best approximation for the ground-state energy.

Optimization and Binding Energy

Optimization in quantum variational calculations involves adjusting the parameters in the trial wavefunction to achieve the lowest possible energy expectation value. The resulting minimum energy can be interpreted as the binding energy of the system, which indicates how strongly particles (such as electrons and nuclei) are bound together. A deeper binding energy generally signifies a more stable and tightly bound system.

Variational Principle

The variational principle is a fundamental method in quantum mechanics used to estimate the ground-state energy of a system. It involves choosing a trial wavefunction with one or more adjustable parameters and then optimizing these parameters so that the expectation value of the Hamiltonian is minimized. This minimum value provides an upper bound to the true ground-state energy, making the variational method a powerful tool for systems where exact solutions are not available.

Trial Wavefunctions

Trial wavefunctions are approximate wavefunctions chosen to capture the essential features of a quantum system. In the variational method, these wavefunctions depend on one or more adjustable parameters that are tuned to minimize the energy expectation value. Using a functional form, such as an exponential decay governed by a variational length scale, allows for a flexible yet tractable model of the quantum state.

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