Take as a trial function for the ground state of the hydrogen atom (a) e\(k r\), (b) \(\mathrm{e}^{-k r}\) and use the variation principle to find the optimum value of \(k\) in each case. Identify the better wavefunction. The only part of the laplacian that need be considered is the part that involves radial derivatives (eqn 9.5).

In quantum mechanics, the 'trial wavefunction' is a guess or an approximation used to calculate properties of particles, such as electrons in an atom. It plays a central role in the 'Variational Principle', which is a method used to estimate the ground state energy of a system. The trial wavefunction is denoted by \( \psi(\text{trial}) \) and it should reasonably comply with the boundary conditions of the actual wavefunction. However, it often contains undetermined parameters that can be adjusted to make the trial wavefunction as close as possible to the actual wavefunction.

For instance, in solving for the hydrogen atom's ground state energy, one might use trial wavefunctions like \( \psi(a) = e^{kr} \) or \( \psi(b) = e^{-kr} \) where \( k \) is a parameter that we adjust to minimize the expected energy. The key to selecting an efficient trial wavefunction is to ensure it has the right symmetry and behavior at the appropriate limits (\( r \rightarrow 0 \) and \( r \rightarrow \infty \) for an atom).

The 'ground state energy' in quantum mechanics is the lowest possible energy that a quantum system may have. For the hydrogen atom, we are generally interested in finding this minimal energy, denoted as \( E_0 \). The variational principle provides a valuable tool for this purpose, as it states that the expectation value of the Hamiltonian, given any trial wave function, will always be greater than or equal to the true ground state energy.

Using the variational method, we compute the Hamiltonian expectation value for different trial wavefunctions and adjust our parameters to find the minimum possible energy. This energy is the best estimate for the ground state energy given the constraints and form of the trial wavefunction. Since the true ground state energy of the hydrogen atom is known from Schrödinger's equation, in practice, this method helps to check the effectiveness of different trial wavefunctions.

The 'Hamiltonian expectation value' is a fundamental concept in quantum mechanics that represents the average energy for a particle or system described by a given wavefunction. Mathematically, it is denoted as \( \langle H \rangle \) and is found by integrating the product of the trial wavefunction, the Hamiltonian operator, and the conjugate of the trial wavefunction over all space.

The expectation value provides a method to calculate the energy levels of a quantum system using trial wavefunctions. It is crucial in the variational principle to determine the lowest energy state, or ground state, of the system. Calculating this expectation value for different trial wavefunctions and varying parameters, we can estimate the ground state energy of a system by finding the minimum value of \( \langle H \rangle \) as it approaches closer to the true energy of the system.

The 'hydrogen atom wavefunction' describes the probability amplitude for the position of an electron in the simplest atomic system: the hydrogen atom. The exact wavefunctions for the hydrogen atom are solutions to the Schrödinger equation with a Coulomb potential due to the positively charged nucleus. The true ground state wavefunction for hydrogen is denoted as \( \psi_{100} \) and is characterized by the absence of nodes -- it has a spherical symmetry around the nucleus.

In the context of the variational principle, a chosen trial wavefunction, such as \( \psi(b) = e^{-kr} \) from the textbook example, is an approximation to this true wavefunction. Through optimization of the parameter \( k \), we seek to find the best approximation which will yield the lowest Hamiltonian expectation value, thus estimating the ground state energy and simulating the behavior of the actual hydrogen atom wavefunction.