Hydrogen atom
THE HYDROGEN ATOM
There are several introductory problems that can be solved exactly by quantum mechanical methods.
These include the particle in a one-dimensional box, the particle in a three-dimensional box, the
rigid rotor, the harmonic oscillator, and barrier penetration. All of these models provide additional
insight into the methods of quantum mechanics, and the interested reader should consult a quantum
mechanics text such as those listed in the references at the end of this chapter. Because of the nature , we will progress directly to the problem of the hydrogen atom, which was solved in 1926 byErwin Schrödinger. His starting point was a three-dimensional wave equation that had been developed
earlier by physicists who were dealing with the so-called fl ooded planet problem. In this model, a
sphere was assumed to be covered with water, and the problem was to deal with the wave motion that
would result if the surface were disturbed. Schrödinger did not derive a wave equation. He adapted
one that already existed. His adaptation consisted of representing the wave motion of an electron by
means of the de Broglie relationship that had been established only 2 years earlier. Physics was progressingat a rapid pace in that time period.
We can begin directly by writing the equation
equation
Hˆψ=Eψ (2.35)
and then determine the correct form for the Hamiltonian operator. We will assume that the nucleus
remains stationary with the electron revolving around it (known as the Born-Oppenheimer approximation)
and deal with only the motion of the electron. The electron has a kinetic energy of (1/2) mv 2 ,
which can be written as p 2 /2 m . Equation (2.34) shows the operator for kinetic energy.
The interaction between an electron and a nucleus in a hydrogen atom gives rise to a potential energy
that can be described by the relationship = e 2 / r . Therefore, using the Hamiltonian operator and postulate
IV, the wave equation can be written as to separate the variables. To circumvent this problem, a change of coordinates to polar coordinates is
made. After that is done, the Laplacian must be transformed into polar coordinates, which is a tedious
task. When the transformation is made, the variables can be separated so that three second-order differential equations, each containing one coordinate as the variable, are obtained. Even after this is done,the resulting equations are quite complex, and the solution of two of the three equations requires the use of series techniques. The solutions are described in detail in most quantum mechanics books, soit is not necessary to solve the equations .These wave functions are referred to as hydrogen-like wave functions
because they apply to any one-electron system (e.g., He , Li 2 ).
From the mathematical restrictions on the solution of the equations comes a set of constraints known
as quantum numbers . The fi rst of these is n , the principal quantum number, which is restricted to integer
values (1, 2, 3, … ). The second quantum number is l , the orbital angular momentum quantum
number, and it must also be an integer such that it can be at most ( n 1). The third quantum number
is m , the magnetic quantum number, which gives the projection of the l vector on the z axis as shown
m , ±1,±2,…,±l .
Note that from the solution of a problem involving three dimensions, three quantum numbers result,
unlike the Bohr approach, which specifi ed only one. The quantum number n is essentially equivalent
to the n that was assumed in the Bohr model of hydrogen.
A spinning electron also has a spin quantum number that is expressed as 1/2 in units of . However,
that quantum number does not arise from the solution of a differential equation in Schrödinger’s
solution of the hydrogen atom problem. It arises because, like other fundamental particles, the electronhas an intrinsic spin that is half integer in units of , the quantum of angular momentum. As a
result, four quantum numbers are required to completely specify the state of the electron in an atom.
The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of four quantum numbers . We will illustrate this principle later.
The lowest energy state is that characterized by n = 1, which requires that l = 0 and m = 0. A state for
which l = 0 is designated as an s state so the lowest energy state is known as the 1 s state, since states
are designated by the value of n followed by a lower case letter to represent the l value. The values of l
are denoted by letters as follows :Value of l : 0 1 2 3
State designation: S harp p rincipal d iff use f undamental
Modeling
the Hydrogen Atom
In 1914, Niels Bohr obtained the
spectral frequencies
of the hydrogen atom after making a number of simplifying assumptions. These
assumptions, the cornerstones of the Bohr model, were not fully correct but did
yield the
correct energy answers. Bohr's results for the
frequencies and underlying energy values were confirmed by the full
quantum-mechanical analysis which uses the Schrödinger equation, as was shown
in 1925–1926. The solution to the Schrödinger equation for hydrogen is
analytical. From this, the hydrogen energy levels and thus the frequencies of
the hydrogen spectral lines can be calculated. The solution of the Schrödinger
equation goes much further than the Bohr model, because it also yields the
shape of the electron's wave function (orbital) for the various possible
quantum-mechanical states, thus explaining the anisotropic
character of atomic bonds.
These include the particle in a one-dimensional box, the particle in a three-dimensional box, the
rigid rotor, the harmonic oscillator, and barrier penetration. All of these models provide additional
insight into the methods of quantum mechanics, and the interested reader should consult a quantum
mechanics text such as those listed in the references at the end of this chapter. Because of the nature , we will progress directly to the problem of the hydrogen atom, which was solved in 1926 byErwin Schrödinger. His starting point was a three-dimensional wave equation that had been developed
earlier by physicists who were dealing with the so-called fl ooded planet problem. In this model, a
sphere was assumed to be covered with water, and the problem was to deal with the wave motion that
would result if the surface were disturbed. Schrödinger did not derive a wave equation. He adapted
one that already existed. His adaptation consisted of representing the wave motion of an electron by
means of the de Broglie relationship that had been established only 2 years earlier. Physics was progressingat a rapid pace in that time period.
We can begin directly by writing the equation
equation
Hˆψ=Eψ (2.35)
and then determine the correct form for the Hamiltonian operator. We will assume that the nucleus
remains stationary with the electron revolving around it (known as the Born-Oppenheimer approximation)
and deal with only the motion of the electron. The electron has a kinetic energy of (1/2) mv 2 ,
which can be written as p 2 /2 m . Equation (2.34) shows the operator for kinetic energy.
The interaction between an electron and a nucleus in a hydrogen atom gives rise to a potential energy
that can be described by the relationship = e 2 / r . Therefore, using the Hamiltonian operator and postulate
IV, the wave equation can be written as to separate the variables. To circumvent this problem, a change of coordinates to polar coordinates is
made. After that is done, the Laplacian must be transformed into polar coordinates, which is a tedious
task. When the transformation is made, the variables can be separated so that three second-order differential equations, each containing one coordinate as the variable, are obtained. Even after this is done,the resulting equations are quite complex, and the solution of two of the three equations requires the use of series techniques. The solutions are described in detail in most quantum mechanics books, soit is not necessary to solve the equations .These wave functions are referred to as hydrogen-like wave functions
because they apply to any one-electron system (e.g., He , Li 2 ).
From the mathematical restrictions on the solution of the equations comes a set of constraints known
as quantum numbers . The fi rst of these is n , the principal quantum number, which is restricted to integer
values (1, 2, 3, … ). The second quantum number is l , the orbital angular momentum quantum
number, and it must also be an integer such that it can be at most ( n 1). The third quantum number
is m , the magnetic quantum number, which gives the projection of the l vector on the z axis as shown
m , ±1,±2,…,±l .
Note that from the solution of a problem involving three dimensions, three quantum numbers result,
unlike the Bohr approach, which specifi ed only one. The quantum number n is essentially equivalent
to the n that was assumed in the Bohr model of hydrogen.
A spinning electron also has a spin quantum number that is expressed as 1/2 in units of . However,
that quantum number does not arise from the solution of a differential equation in Schrödinger’s
solution of the hydrogen atom problem. It arises because, like other fundamental particles, the electronhas an intrinsic spin that is half integer in units of , the quantum of angular momentum. As a
result, four quantum numbers are required to completely specify the state of the electron in an atom.
The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of four quantum numbers . We will illustrate this principle later.
The lowest energy state is that characterized by n = 1, which requires that l = 0 and m = 0. A state for
which l = 0 is designated as an s state so the lowest energy state is known as the 1 s state, since states
are designated by the value of n followed by a lower case letter to represent the l value. The values of l
are denoted by letters as follows :Value of l : 0 1 2 3
State designation: S harp p rincipal d iff use f undamental
References
· Christian Laurence and Jean-François Gal "Lewis Basicity and Affinity Scales : Data and Measurement" Wiley, 2009.
· Lewis, G.N., Valence and the Structure of Atoms and Molecules (1923) p. 142.
· Miessler, L. M., Tar, D. A., (1991) p166 - Table of discoveries attributes the date of publication/release for the Lewis theory as 1923.
March, J. “Advanced Organic Chemistry” 4th Ed. J. Wiley and Sons, 1992: New York
· Greenwood, N. N.; & Earnshaw, A. (1997). Chemistry of the Elements (2nd Edn.), Oxford:Butterworth-Heinemann.
· Jensen, W.B. (1980). The Lewis acid-base concepts : an overview. New York:.
· Yamamoto, Hisashi (1999). Lewis acid reagents : a practical approach. New York: Oxford University Press.
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