Introduction
 
The hydrogen atom has a long and rich history in quantum mechanics, extending back to Niels Bohr and his radical model of a planetary hydrogen atom whose electron could only exist at specific energy levels.  The hydrogen atom is the most basic element of the periodic table, with only one electron orbiting a nucleus that consists of one proton.  Due to the hydrogen atom’s simplicity, it serves to be an incredibly useful model in quantum mechanics, used in predictions and modeling quantum experiments. The Schrodinger equation is a fundamental concept in quantum mechanics because it models how the state of a quantum system changes over time.  Since the Schrodinger equation can be solved exactly for hydrogen and other two-body models, as interactions between multiple electrons result in complications for many-body models, the significance of hydrogen in quantum chemistry is huge.

 

N-Body Problem

 

The motion of particles interacting with each other can be described by solving an n-body problem, where n is the number of particles involved. In quantum mechanics, a hydrogen atom is used as a simple model because it has two bodies, a proton and an electron. Solving a 2-body problem is much simpler than solving a many-body problem; in fact, it is impossible to solve a 3-or-more-body problem exactly (Faddeev 1993). The state of the hydrogen atom can be found by solving a mathematical equation that describes the system.

 

 

A many-body problem is difficult because of its many variables. Adding an electron to the proton-electron system would introduce an electron-electron repulsion term, making the equation impossible to solve. There are methods for approximating higher-body problems, but they are not true solutions. The Hartree method, for example, treats the three-body problem of two electrons and a nucleus as a two-body problem by replacing one of the electrons with a potential energy field. A hydrogen atom, or a hydrogen-like atom (with one nucleus and one electron, such as He⁺), can be solved completely; due to its simplicity, scientists have been able to take important steps in quantum mechanics.

 

Because the mass of a proton is much greater than that of an electron, the two bodies in the hydrogen atom can be treated as a moving electron and a stationary proton (McQuarrie 2008). Mathematically, the function that describes the motion of the electron is called its wavefunction (discussed in detail below). The wavefunction, the probability distributions of the position and momentum of the electron, and the energy of the system can all be found by solving Schrodinger’s equation, Hψ(x)=Eψ(x) (also discussed below).

 

Wavefunction

 

Once electrons were found to exhibit wavelike properties, electrons could be modeled analogous to the functions that model classical waves, in the sense that the second derivatives of these functions produce the original function multiplied by a constant. Thus, the similarities lie only within the realm of mathematics. A wavefunction for an electron, which is represented by Ψ (psi), is more complex than a function for a classical wave; it contains all observable measurements of a particle, such as its momentum or potential energy. However, this information in its current form, Ψ (psi), does not provide any information on the position of a particle, unlike a wavefunction for a classical wave. In order to convert Ψ into something more usable, we must alter it first. 

 

 

Figure 1: Wavefunctions for Particle-In-A-Box, where an electron is trapped between two "walls," on the left and for the Hydrogen atom on the right. Note how the appearances of a wavefunction vary depending on the constraints and the environments of the systems being studied.

 

Once a wavefunction is multiplied by its complex conjugate, which is called Ψ* (psi star), (see Figure 2 for an example of this process) the resulting product, Ψ², is a function that describes the probability of finding an electron within a certain 3D space (a fancy way of saying volume). Once again, there are certain restrictions. When the wavefunction is multiplied by its complex conjugate, the resulting function must be normalized, meaning the total area between the function and the x axis is equal to one and the limit of the function as it approaches positive or negative infinity is zero. To do this, the integral of Ψ² is taken from negative to positive infinity. This integral is then multiplied by a constant squared (in this case A²) and the overall integral is set to equal one (see Equation 1). Thus, A² is the constant that is multiplied with Ψ² to normalize it. Note that A can be used to normalize psi and psi star in a similar manner.

 

Moreover, wavefunctions can be in multiple dimensions, in the example below, we consider normalizing a wavefunction with one dimension for the sake of simplicity. To normalize a wavefunction with more than one dimension, set each limit of integration to the maximum and minimum values that said limit of integration can attain. For example, to normalize the triple integral a few paragraphs below this (Equation 2), we follow the same procedure as above except that the limits of integration for θ is zero to 180 degrees, for ϕ is zero to 360 degrees, for r is zero to positive infinity.

 

Equation 1: Normalizing Ψ² 

Figure 2: Complex Conjugates

Note, that in this diagram, j represents a complex number. Thus, a(x) represents the real part of Ψ and jb(x) represents the imaginary part of Ψ. Thus, a complex conjugate simply has a sign change on its imaginary part.

 

Moreover, when a wavefunction is multiplied by its complex conjugate, and provided that the resulting function is normalized, the function produced also describes the distribution of electron positions. Thus, in the case of hydrogen, given any position of an electron relative to the nucleus, Ψ² can find the probability of such a position within a certain space (in other words, a certain volume). These distributions determine the shapes of an atom's orbitals, which are regions where there is a certain probability of finding an electron (ex. 90%).  This probability is called a probability density, since it relates probability per unit of volume. In other words, it is the probability of finding an electron within a certain 3D area.

 

To find these probability densities, we integrate Ψ², making sure to first multiply the integral by A² to normalize it, over the volume of interest. We use integration since the probability of finding an electron decreases as we move away from the nucleus of an atom. This makes sense conceptually since a nucleus is positively charged and an electron is negatively charged. Both particles are attracted to each other and it makes sense that they would most likely be "close" together. Thus, as we vary our position relative to the nucleus, the probability density of finding an electron varies as well. This sounds complicated, but what we just described is Ψ², it provides a probability density for any position of the electron, however, there's more to finding probability density than just that. Since we are looking for the probability of finding an electron within a certain volume, we break up this volume into very small chunks. For every small chunk of volume, we find the probability density for it and add each of these probability densities together to find the overall probability density for the overall volume. This sounds like tedious work, but integration takes care of that for us in one fell swoop. We take Ψ² and integrate it over the volume that we wish to study to obtain the corresponding probability density.

 

Before we begin integrating anything, it is important to consider the geometry of the interactions between an electron and the nucleus of a hydrogen atom. The interaction between an electron and the nucleus of a Hydrogen atom is dependent only on the distance between the two and not on their angular orientations. In simpler terms, the "region of interest," or the volume that we wish to study, is really the volume of a sphere. Using Cartesian coordinates (x, y, and z) to integrate Ψ² is not very well suited for finding the volumes of spherical objects. Thus, we use spherical coordinates instead, which are outlined below.

Equation 2: Where r²sin(θ)drdθdϕ = dxdydz and r is the distance of the electron from the origin, θ is the angle between the z-axis and the electron, and ϕ is the angle between the x-axis and the electron when the electron is projected onto the x-y plane (see figure below).

 

 

Note that A² is the normalizing factor and the triple integral behind it represents the volume of a sphere that is described by our limits of integration. Moreover, we can also find the distance from the nucleus that corresponds to a certain probability density, for instance, if we wanted to find the distance from the nucleus that would have a 90% probability density, then we simply set the above integral, the triple integral with spherical coordinates, equal to 0.90 (since the integral is normalized, it equals 1, which represents the volume that has a probability density of 100%, and 90% of 1 is 0.90) and solve for the limits of integration, which determine the geometry of a sphere.

 

It is through this method, that we can find the probability density of an electron in a Hydrogen atom, and other Hydrogen like atoms (ie. atoms with one electron, like He⁺). This approach can be extended to atoms with multiple electrons, but not with the same success. As outlined above in the N-Body Problem, multiple electrons add complexities to our system, due to added forces such as electron-electron repulsion. Not only will multiple wavefunctions, one for each electron, come into play, but we must also account for electron-electron repulsions, which can vary in magnitude and direction depending on the relative positions between each electron and the directions said electrons come into contact with each other. Unlike the varying probability density between a Hydrogen nucleus and its electron, the electron-electron repulsion of multi-electron atoms cannot be measured as succinctly. Electrons in a multi-electron atom are in constant motion and can interact with each other from any direction. When more and more electrons are considered, the complicating powers of electron-electron repulsion becomes more and more aggravated. Thus, the beauty of Hydrogen-like atoms is that they do not have electron-electron repulsion, and can be solved quite simply, while multi-electron atoms are just the opposite.Despite this, the general method we outlined above can be used to find probability densities of atoms with multiple electrons, except that there are no elegant solutions, as there are for Hydrogen-like atoms, and brute-forcing these solutions will take an extraordinary amount of time combined with extraordinary amounts of computing power. It is not that these "solutions" do not exist, the math that must be dealt with is too much to handle at this point in time.

 

Quantum numbers

 

Using hydrogen to solve the Schrodinger equation results in a wave function, which describes the orbitals where an electron around an atom is confined to. This possible area around an atom where the electron can be found is represented by quantum numbers. Since the electron’s location is found in a three-dimensional space around the Hydrogen atom, it is natural to use 3 quantum numbers to define the three dimensions. Each of these can be found by separating the Schrodinger equation for hydrogen.

 

1. Primary quantum number (n) –represents main energy level that the electron is located. The value of n is equivalent to the shell of the atom that the electron is located, so it can only be an integer value eg. N=1,2,3 etc. A higher number also corresponds to a larger radius of the electron cloud, since the electron would be further away from the nucleus. It is found by solving the radial part of the Schrodinger equation for hydrogen:

Thus, this value of n can only be an integer value.

 

2. Orbital Angular momentum (l) – determines magnitude of angular momentum. This value corresponds to the sublevels (l­₀ = s, l₁ = p, l₂ = d, etc.) and corresponds with m_l to form the shape of the orbital. 

 

 

It is found by solving the colatitude equation of the Schrodinger equation for hydrogen:

Constraints on this equation produces the orbital quantum number:

3. Directional or Magnetic Angular momentum (m­_l) – affects the shape of electron cloud. Can only be –l up to +l. It is found by examining the constraint of the azimuthal equation for the Schrodinger equation for hydrogen:

These three quantum numbers interact to define an orbital around the nucleus where an electron can be found. Each electron has its own unique set of quantum numbers.

 

In this diagram, the primary quantum number corresponds with the spherical coordinate, radius (r), the orbital angular momentum corresponds with angle off the z axis (θ) and the magnetic angular momentum corresponds the angle in the xy- plane (φ)

 

These quantum numbers will appear in the wave function of the electron in question:

A fourth quantum number Electron Spin can be used to describe the spin orientation of individual electrons in an orbital. Since only two electrons are present in any given orbital and these electrons have opposite spin axis, a spin value of +1/2 or -1/2 is given to distinguish the electrons of an orbital from each other. 

 

The Hamiltonian

 

The Hamiltonian is the name for a specific operator that acts on a wave function,, in order to give the total energy of the system described by the wave function.  To hit the basics, first of all, an operator can be any symbol corresponding to a specific mathematical rule that, when you see it, tells you how to evaluate an expression.  Some examples of common operators include the multiplication symbol and the division symbol.  Once we know what an operator is, we might wonder where it came from.  So, to investigate what the Hamiltonian of a hydrogen atom is, we first look at the origins of the Hamiltonian operator.

 

Although we commonly talk about the Hamiltonian as applied to quantum mechanical systems, the concept also exists in classical mechanic systems as well.  As an energy operator, in classical mechanics, the Hamiltonian also gives the total energy of the system, which equals the kinetic energy, T, plus the potential energy, V:

Kinetic energy is equal to (mv²)/2, but this term is commonly expressed in terms of momentum, p=mv, with the equivalent expression, p²/2m:

 

Returning to quantum mechanics, the Hamiltonian is the same expression as the classical equation given above, except now the classical dynamic terms must be represented as quantum operators:

And rewriting the kinetic energy term:

The momentum operator is equal to –ih(d/dx) so plugging it into the quantum Hamiltonian:

 

The wave function for a particle moving in one dimension, like the particle in a box model, is given by the equation:

The first derivative of this wave function is:

And the second derivative is:

By plugging the original wave function equation into the second derivative, we get the equivalent:

Then, according to de Broglie’s equation, where h is Planck’s constant, 6.626 x 10^-34m^2*kg/s and p is momentum:

And taking the equation for kinetic energy, E_k = ½ mv², where  is mass and  is velocity, together with the momentum equation, so that p² = m²v², so that E_k = (p^2)/2m  and p² = 2mE_k:

 

And isolating kinetic energy, E_k, to one side of the equation:

Which gives us the kinetic energy term for the wave function.

 

Note: Alternately, we could rewrite with  for simplicity:

 

Then, to find the total energy of the wave function, , add in the potential energy term V(x) to the kinetic energy term:

As stated at the beginning, the Hamiltonian is a specific operator that acts on wave function in order to give the total energy of that wave, so rewriting the equation to act on the wave function :

We find  to be the operator that acts on the wave function  to give you total energy, and this term is called the Hamiltonian operator.

And, plugging  into the above equation, we can rewrite:

This famous equation is known as Schrödinger’s Equation.

 

Now, we turn our attention to evaluating the Hamiltonian of the hydrogen atom.  Hydrogen is the simplest problem to consider because we need only calculate the energies for two bodies – the kinetic energy of its proton, the kinetic energy of its electron, and the potential energy between them.  Therefore, the Hamiltonian will involve two kinetic energy terms, one for the proton and one for the electron, and the potential energy function is given by Coulomb’s force law for a pair of charged particles. 

 

Thus, the Hamiltonian for hydrogen is as given below, where ∇ is a differential operator - like d/dx except in multiple dimensions, m_p is the mass of a proton and m_e is the mass of an electron, q_p is the charge of a proton, and q_e is the charge of an electron, and r is the distance between the proton and electron:

Questions

 

 

1. Can a wavefunction tell you exactly where an electron is?

 

2. What else besides Hydrogen can be solved exactly by the Schrodinger equation?

 

3. What is the Hamiltonian operator used for?

 

Answers

 

1. No, a wave function cannot tell you exactly where an electron is.  However, it can tell you the probability of finding an electron within a certain 3D space, which is the probability density.

 

2. Other 2 body systems like He⁺ or Li²⁺ can also be solved exactly by the Schrodinger equation.

 

3. The Hamiltonian operator is used to find the total energy of a wave function.

 

 References

 

Asher, Peres, Quantum Theory, Concepts and Methods, Kluwer, 1993

 

Horodecki R, Horodecki P, Horodecki M, Horodecki K (2007). "Quantum entanglement". Rev. Mod. Phys. 81 (2): 865–942.

 

L.D. Faddeev, S.P. Merkuriev. "Quantum Scattering Theory for Several Particle Systems." Springer. August 31, 1993.

 

McQuarrie, Donald A. Quantum Chemistry. 2nd ed. United States Of America: University Science Books, 2008. 321-24.

Micu, L. "On The Two-Body Problem In Quantum Mechanics." Rom. Journ. Phys. 53.9-10 (2008): 1171-176. Print.

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Nave, C. R. "The Hamiltonian." Quantum Mechanics. Georgia State University, n.d. Web. 20 Sept. 2

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Oxtoby, David W., H. P. Gillis, and Alan Campion. Principles of Modern Chemistry. 7th ed. Belmont, CA: Brooks/Cole Cengage Learning, 2012. Print.

 

"Quantum Mechanical H Atom." Chemwiki.ucdavis.edu. N.p., June-July 2010. Web. Sept.- Oct. 2012.

 

Sension, Roseanne. "Chem 260/261 - Lecture 6: Quantum Laws." Chem 260/261 Lecture. University of Michigan, Ann Arbor. 16 Sept. 2012. Lecture.

 

Telfer, Randy. "Schrodinger Theory." Schrodinger Theory. N.p., 26 Oct. 1998. Web. 06 Oct. 2012. <http://www.pha.jhu.edu/~rt19/hydro/node3.html>.

 

 

"The Hydrogen Atom." Hydrogen Schrodinger Equation. N.p., n.d. Web. 01 Oct. 2012. <http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html>.

Van Dommelen, Leon. "The Hydrogen Atom." The Hydrogen Atom. Florida State University, n.d. Web. 20 Sept. 2012.