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. In Chapter 6 it was shown that every one-dimensional...

Key Concepts

Variational Principle

The variational principle is a fundamental method in quantum mechanics used to estimate the ground state energy of a system. By choosing a trial wave function that satisfies the necessary boundary conditions, one computes the expectation value of the Hamiltonian, which is guaranteed to be an upper bound to the true ground state energy. This principle is rigorously justified and provides a systematic way to prove the existence of bound states, as any trial state with an energy expectation below a certain threshold implies that a true bound state exists with energy no greater than that value.

Bound States

A bound state in quantum mechanics refers to a state in which the particle is confined to a finite region of space, characterized by a normalizable wave function and an energy lower than the potential at infinity. Bound states typically arise in potentials that have a well-like shape, enabling the existence of discrete energy levels. The presence of at least one bound state in a given potential has far-reaching implications on the stability and spectral properties of the system.

One-Dimensional Potentials

One-dimensional potentials are functions that depend solely on one spatial coordinate and define the energy landscape in which a quantum particle moves. In the context of the variational principle, certain properties of these potentials, such as being negative everywhere and asymptotically approaching zero at infinity, can be critical. These characteristics ensure that the potential forms a well deep enough to support bound states, even if the depth or width of the well varies.

Trial Wave Function

A trial wave function is a chosen function that approximates the actual wave function of a quantum system and satisfies the necessary boundary and normalizability conditions. In the variational method, this function is used to calculate an upper bound to the ground state energy. An appropriate choice of a trial wave function, especially in systems where the potential is negative everywhere, can demonstrate that the expectation value of the energy falls below zero, thereby proving the existence of a bound state.

Expectation Value of Energy

The expectation value of energy is computed by taking the integral of the product of the complex conjugate of the trial wave function, the Hamiltonian operator, and the trial wave function itself, divided by the normalization integral of the wave function. This value provides an estimate of the energy that a quantum state would have. When the variationally computed expectation value of energy is found to be less than the potential energy at infinity, it confirms that the system must possess a bound state with energy below that asymptotic level.

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