The Hydrogen Atom As Viewed By Quantum Mechanics (1963)

here is the filament in a Tungsten lamp as the filament is heated it emits energy in the form of light although the light appears white to our eyes when we pass the light through a prism it is separated into a continuous spectrum of colors this continuous Spectrum shows that a Tungsten lamp emits photons of all energies this is a gas discharge tube containing hydrogen the light emitted by this tube is due to hydrogen atoms let's look at the Spectrum of this light source this time we see a line Spectrum each line corresponds to just one possible energy change in a hydrogen atom we see that very few energy changes occur in hydrogen atoms in contrast to the tungsten filament accurate frequency measurements tell us the energy changes and lead us to the hydrogen atom energy level [Music] diagram but how can we explain the fact that the energy of a hydrogen atom has only these certain values we begin with a simple picture of the hydrogen atom at any given instant the electron is separated from the proton by some distance R the electron and the nucleus have opposite electric charges so they attract each other this attraction gives the atom a certain amount of potential energy the amount depends on the distance R the electron is in motion and so the atom also has kinetic energy the sum of these two kinds of energy is the total electronic energy of the atom this sum expresses the laws of motion that apply to macroscopic particles to fit The observed behavior of microscopic particles like electrons and protons the laws must be modified somewhat as modified these laws are called quantum mechanics the mathematics of quantum mechanics are quite complex but we don't need to consider the details of the calculations our interest is in the results that quantum mechanics gives this high-speed computer simplifies the task TK of carrying out the calculations the equation is punched onto cards and fed into the computer together with instructions on how to solve it the computer then rapidly solves a problem that is tedious if done by hand when the calculations are completed the machine types out the energy values let's compare these calculated energies with the experimentally determined energies of the hydrogen atom agreement is excellent thus we see that quantum mechanics provides an explanation for the specific energies observed for the hydrogen atom because quantum mechanics also confirms all other properties of the hydrogen atom that can be measured we believe the results quantum mechanics gives for those properties that are more difficult to measure one such property is the location of the electron relative to the nucleus what can we learn from quantum mechanics about this to answer this question we will consider the 1s State the lowest and most stable energy state of the hydrogen atom quantum mechanics gives the results only in terms of probability a dartboard will help us understand the kind of probability information that we get from quantum mechanics see where the dart hits each throw is completely independent of the others observe the pattern of punctures there are many holes concentrated near the bullseye toward the edge of the board there are fewer punctures these holes indicate the likelihood that the next throw of the dart will land in a given square cm where there are many holes the probability is high where there are few holes the probability is low these observations allow us to conclude that the probability of the dart landing near the bullseye is much greater than the probability of it Landing far away we shall see that there is a close analogy to the electron distribution in the hydrogen atom the pattern of holes is centered around the bullseye just as the electron centers its movement around the nucleus compare it to the probability plot for the 1s state of the hydrogen at as generated by the computer what does this plot tell us about the location of the electron to answer this question another analogy may be useful the movements of a hummingbird around his nest we'll use the hummingbird to represent the electron his nest represents the nucleus The Nest is located in the small tree in a field of flowers let's represent this scene on a chart here is his nest in the tree the hummingbird collects nectar from the flowers here and there around his nest let's suppose that we keep a record on this map of the location of each flower the hummingbird visits here's one now we'll mark the location of this flower on the map oh now he's over here oops there he goes let's mark it ah and now a third let's continue plotting these locations plotting over a long period of time a pattern develops in this pattern for a given area the number of pins is much higher near the nest than it is farther away again we have probability information thus we conclude that the probability of the bird landing on a flower near the nest is much greater than the probability of landing on a flower some distance away we can discuss the average distance of the hummingbird from The Nest but sometimes he will be found closer than this average distance sometimes farther away in any event we see that there are no boundaries to the pattern occasionally the hummingbird lands on a flower a very great distance away the plot of the hydrogen atom which the computer Drew earlier has exactly the same meaning just as this map tells us the probability of finding the hummingbird a given distance from his nest so this plot tells us the probability of finding the electron a given distance from the proton as in our analoges we see that the probability of finding the electron near the nucleus is much higher than finding it far away again we find that there are no boundaries to this plot for although the electron is most likely to be found near the nucleus it occasionally may be found very far away a limitation of this plot is that it chose probability in only two Dimensions we can use animation to add depth to the plot this probability description of electron location is called the 1s orbital of the hydrogen atom we note that the orbital has no definite boundaries although it does have a spherical shape thus the expression 1s is a convenient abbreviation where the one indicates the energy state of the atom and the S denotes the spherical shape of the orbital but remember that the orbital represents only the probability of finding the electron at a certain location if we could observe the electron at any one instant the electron would be found at some particular location at another instant it would be found at some other location what about the path or trajectory that the electron follows in getting from one location to another well this is a question that quantum mechanics does not answer explicitly any trajectory that corresponds to the spatial distribution we see here could be in accord with quantum mechanics thus we can only test trajectory models of the atom by comparing them to the quantum mechanical picture for example here is a common model of the electron trajectory that you have probably seen often it shows the electron moving around the nucleus in a path similar to that of a planet moving around the Sun we'll call this a satellite model let's compare the satellite model to what we have learned from quantum mechanics concerning the hydrogen atom if the satellite model were correct the electron would always be at a fixed distance from the nucleus then the probability plot would take the form of a circle the electron would never come very close to the nucleus but experiments show that the electron does come close to the nucleus just as calculated with quantum mechanics hence the satellite model is inconsistent with our experimental knowledge of atoms and can be discarded although quantum mechanics does not tell us the detailed motion of the electron in the atom it does tell us the electron position in terms of probability far from the nucleus it is low near the nucleus it's high we can consider an average radius of the electron position but there are no boundaries to the atom the orbital contains only probability information just like the pattern of punctures in a dart board just like the map of the flowers visited by the hummingbird this plot determined by quantum mechanics describes the structure of the hydrogen atom in terms of probability quantum mechanics also explains many other concepts of chemistry such as is the energy levels of many electron atoms the formation of molecules bond energies and bond lengths these successes prove the usefulness of the quantum mechanical view of the atom