Physics
Hydrogen Wave Function
The hydrogen wave function describes the probability amplitude of finding a hydrogen atom's electron at a particular location in space. It is a solution to the Schrödinger equation for the hydrogen atom and provides information about the electron's energy levels and spatial distribution. The wave function is used to calculate various properties of the atom, such as its orbital shapes and sizes.
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11 Key excerpts on "Hydrogen Wave Function"
eBook - PDF- Kenneth S. Krane(Author)
- 2020(Publication Date)
- Wiley(Publisher)
Chapter 7 THE HYDROGEN ATOM IN WAVE MECHANICS These computer-generated distributions represent the probability to locate the electron in the n = 8 state of hydrogen for angular momentum quantum number l = 2 (top) and l = 6 (bottom). The nucleus is at the center, and the height at any point gives the probability to find the electron in a small volume element at that location in the xz plane. This way of describing the motion of an electron in hydrogen is very different from the circular orbits of the Bohr model. © John Wiley & Sons, Inc. 208 Chapter 7 The Hydrogen Atom in Wave Mechanics In this chapter, we study the solutions of the Schrödinger equation for the hydrogen atom. We will see that these solutions, which lead to the same energy levels calculated in the Bohr model, differ from the Bohr model by allowing for the uncertainty in localizing the electron. Other deficiencies of the Bohr model are not so easily eliminated by solving the Schrödinger equation. First, the so-called “fine structure” of the spectral lines (the splitting of the lines into close-lying doublets) cannot be explained by our solutions; the proper explanation of this effect requires the introduction of a new property of the electron, the intrinsic spin. Second, the mathematical difficulties of solving the Schrödinger equation for atoms containing two or more electrons are formidable, so we restrict our discussion in this chapter to one-electron atoms, in order to see how wave mechanics enables us to under- stand some basic atomic properties. In Chapter 8, we discuss the structure of many-electron atoms. 7.1 A ONE-DIMENSIONAL ATOM Quantum mechanics gives us a view of the structure of the hydrogen atom that is very different from the Bohr model. In the Bohr model, the electron moves about the proton in a circular orbit.
eBook - PDFQuantum Dynamics
Applications in Biological and Materials Systems
- Eric R. Bittner(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
2 Waves and Wave Functions In the world of quantum physics, no phenomenon is a phenomenon until it is a recorded phenomenon. John Archibald Wheeler Bohr’s model of the hydrogen atom was successful in that it gave us a radically new way to look at atoms. However, it has serious shortcomings. It could not be used to explain the spectra of He or any multielectron atom. It could not predict the intensities of the H absorption and emission lines. With de Broglie’s hypothesis that matter was also wavelike, 1 there arose a question at the 1925 Solvey conference: What is the wave equation? De Broglie could not answer this; however, over the next year Erwin Schr¨ odinger, working in Vienna, published a series of papers in which he deduced the general form of the equation that bears his name and applied it successfully to the hydrogen atom. 2 , 3 What emerged was a new set of postulates, much like Newton’s, that laid the foundations of quantum theory. The physical basis of quantum mechanics is 1. That matter, such as electrons, always arrives at a point as a discrete chunk, but that the probibility of finding a chunk at a specified position is like the intensity distribution of a wave. 2. The “quantum state” of a system is described by a mathematical object called a “wave function” or state vector and is denoted | ψ . 3. The state | ψ can be expanded in terms of the basis states of a given vector space, {| φ i } , as | ψ = i | φ i φ i | ψ (2.1) where φ i | ψ denotes an inner product of the two vectors. 4. Observable quantities are associated with the expectation value of Hermi-tian operators and that the eigenvalues of such operators are always real. 5. If two operators commute, one can measure the two associated physical quantities simultaneously to arbitrary precision. 6. The result of a physical measurement projects | ψ onto an eigenstate of the associated operator | φ n yielding a measured value of a n with probability | φ n | ψ | 2 . 33
- Sakir Erkoc, Turgay Uzer(Authors)
- 1996(Publication Date)
- World Scientific(Publisher)
Ch.5 Quantum theory of one-electron atoms 99 Fig. 5.4. Radial wave functions for hydrogenic atoms. 100 Lecture notes on atomic and molecular physics; Erkog ni ) at time zero. For a hydrogen-like atom interacting with radiation, the potential energy function includes the terms representing the interaction of the electron with the electromagnetic wave. The most important of these is the electric dipole term. For example, for a wave with the electric field oriented in the x-direction, the electric dipole term is —exE x .
eBook - PDFQuantum Mechanics
A Paradigms Approach
- David H. McIntyre(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
C H A P T E R 8 Hydrogen Atom The angular wave functions we found in the last chapter are independent of the particular form of the central potential that binds the system. The remaining radial part of the wave function, however, depends critically on the central potential you choose. The radial part of the problem determines the allowed energies of the system and hence the spectroscopic fingerprint of the system that we observe in experiments. In this chapter, we solve for the quantized energies and the radial wave functions of the bound states of the hydrogen atom, which is the simplest atomic system, comprising one electron bound to one proton in the nucleus. The electron and proton are bound together by the Coulomb poten- tial, which underlies the bonding in all atoms, molecules, liquids, and solids. 8.1 THE RADIAL EIGENVALUE EQUATION In Chapter 7, we separated the three-dimensional energy eigenvalue equation into differential equa- tions for each of the spherical coordinates r, u, and f. We solved the f eigenvalue equation (7.83) and found the azimuthal eigenstates m 1f2 and eigenvalues m, which determined the separation constant B = m 2 . We then used the separation constant B to make the u differential equation (7.82) into an eigenvalue equation and solved for the polar eigenstates m / 1u2 and the eigenvalues /1/ + 12 , which determined the separation constant A = /1/ + 12 . We now use the separation constant A to make the radial differential equation (7.79) into an eigenvalue equation for the energy E: c - U 2 2mr 2 d dr ar 2 d dr b + V1r2 + /1/ + 12 U 2 2mr 2 d R1r2 = ER1r2. (8.1) Solving this differential equation will give us the radial eigenstates R1r2 and the allowed ener- gies E. We then combine the three separated eigenstates into the three-dimensional eigenstate c1r, u, f2 = R1r2Y m / 1u, f2 , where the spherical harmonics Y m / 1u, f2 = m / 1u2 m 1f2 are the prod- ucts of the azimuthal and polar eigenstates that we found in Chapter 7.
eBook - PDF- Arthur Adamson(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
16-7 SOLUTIONS OF THE WAVE EQUATION FOR THE HYDROGEN ATOM 677 out of Eq. (16-54) and the limiting solution is of the form çxp(±q 2 /2); however, only the minus sign is physically acceptable since otherwise the catastrophe of φ going to infinity would occur. This conclusion then leads to Eqs. (16-56) and (16-57), the latter being Hermite's equation. Only integral values of η are allowed in the solution to Eq. (16-58) if the polynomials satisfying the equation are to have a finite number of terms and not themselves lead to a catastrophe. The quantization thus arises from restriction of the mathematical possibilities to physically acceptable solutions. The most complete application of wave mechanics is undoubtedly to the case of the hydrogen atom. Not only can exact solutions be obtained, but these solutions are then widely used in approximate treatments of heavier atoms. We proceed in this section to the formal solutions for the hydrogen atom and will discuss the behavior of these solutions in detail in the following section. The derivation that follows is more frightening in appearance than actually difficult! A. The Schrbdinger Equation for the Hydrogen Atom The potential function for the electron is now —e 2 /r, where r is its distance from the nucleus, and Eq. (16-29) becomes The presence of r as a parameter in the potential function makes it awkward for one to work in Cartesian coordinates and it turns out to be very convenient to use the polar coordinate system shown in Fig. 16-7. A point at (x, y, z) is now given 16-7 Solutions of the Wave Equation for the Hydrogen Atom (16-67) ζ / / Ύ χ y FIG. 16-7. The polar coordinate system used for the hydrogen atom. 678 CHAPTER 16: WAVE MECHANICS by its radial distance r and the two angles θ and φ. In sweeping out all of space r varies from 0 to oo, θ varies from 0 to π, and φ varies from 0 to 2π.
eBook - ePubTheoretical Spectroscopy of Transition Metal and Rare Earth Ions
From Free State to Crystal Field
- Mikhail G. Brik, Ma Chong-Geng, Mikhail G. Brik, Ma Chong-Geng(Authors)
- 2019(Publication Date)
- Jenny Stanford Publishing(Publisher)
n, l, m determine the expression for the total wave function. Although the general form of the wave functions seems to be rather complicated, the explicit form of the first wave functions corresponding to the lowest energy levels for the atom of hydrogen is relatively simple. As an example, several wave functions are given below (r is measured in units of a0 and Z = 1):Ψ 100=1 πexp(− r)Ψ 200=142 π(exp2 − r)(−)r 2Ψ 210=1r exp42 π(cos θ−)r 2Ψ 211=1r exp8 π(sin θ−)r 2ei ϕΨ=21 − 11r exp8 π(sin θ−)r 2e.− i ϕThe physical meaning of the wave function is as follows: The square of the absolute value of the wave function Ψnlm at a given point is the probability density to find an electron at that point. Figures 3.2 , 3.4 show the radial part of the probability |Ψnlm |2 r2 for several Hydrogen Wave Functions. These plots (obtained with the help of codes from the F. Y. Wang’s book “Physics with Maple” cited at the end of this chapter) in the polar coordinates are the cross sections of the surfaces corresponding to a particular value of probability. The | Ψnlm |2 r2 function has no φ term; in all figures the z-axis in the space corresponds to the horizontal axis in the figure, and the angle θ is measured with respect to the horizontal axis. These plots illustrate that the electron density distribution is quite different in different states.Figure 3.2 Cross section of the electron density distribution in the Ψ321
eBook - PDF- Arther Adamson(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
An intriguing aspect of the statistical interpretation is in the possibility that the supposedly identical set of experiments which give distributions in ρ and in χ may not really be identical. Might there be some as yet unknown variable which is not being controlled and whose variation leads to Sp and Sxl If so, wave mechanics is not a complete description of nature. As may be imagined, the hidden variable question has given rise to lively and as yet unresolved debate. 16-CN-3 Steps beyond Hydrogen-Like Wave Functions Very extensive advances have been made in the direction of obtaining more accurate solutions to the wave equation for multielectron atoms. The detailed methodology of advanced treatments must be left to subsequent elective or graduate courses. We can, however, outline the directions taken and summarize qualitatively the degree of progress made. The difficulty of dealing with multielectron atoms becomes apparent even with the helium atom. The wave equation is now h 2 i7e 2 7e 2 e 2 ~ 8 ^ 7 ^ + W - ( f + f --y Φ = Εφ. (16-110) where the subscripts refer to electrons 1 and 2; that is, the Hamiltonian is [see Eq. (16-30)] h 2 7 P* 7e 2 e 2 Η = - g^- (V + V/) - - ±1-+ ±- . (16-111) oTT'm r x r 2 r 1 2 One now has a second-order partial differential equation in six variables—three coordinates for each electron. The real difficulty is in the e 2 /r 12 term, which gives the interelectronic repulsion. Were it not for this, the wave equation could be separated into two, one for each electron. Put another way, the helium atom is the wave mechanical three-body problem. It has not been solved explicitly, but successive approximation methods aided by high-speed computers have led to solutions which appear to be exact theoretically and which agree well with experiment (with respect to, say, the spectrum of helium).
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
That is, the hydrogen atom has been ionized, meaning that the electron has been removed to a distance so great that the Coulomb force on it from the nucleus is negligible. The atom can be ionized if it absorbs any wave- length shorter than the series limit. The free electron then has only kinetic energy K (= 1 2 mv 2 , assuming a nonrelativistic situation). Quantum Numbers for the Hydrogen Atom Although the energies of the hydrogen atom states can be described by the single quantum number n, the wave functions describing these states require three quantum numbers, corresponding to the three dimensions in which the electron 1060 CHAPTER 39 MORE ABOUT MATTER WAVES 1061 39-5 THE HYDROGEN ATOM A Nonquantized Nonquantized Nonquantized Nonquantized 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 2 3 4 5 6 n ∞ Energy (eV) 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 1 1 2 3 4 5 6 n ∞ Energy (eV) Lyman series Series limit 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 2 3 4 5 6 n ∞ Energy (eV) Balmer series 1 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 Energy (eV) 1 λ λ λ Paschen series Series limit 2 3 4 5 6 n ∞ Series limit These are the lowest six allowed energies of the hydrogen atom. The Balmer series of wavelengths are jumps up from n = 2 (absorption) or down to n = 2 (emission). The Paschen series of wavelengths are jumps up from n = 3 (absorption) or down to n = 3 (emission). The Lyman series of wavelengths are jumps up from n = 1 (absorption) or down to n = 1 (emission). This is the shortest Balmer (series limit). λ This is the longest Balmer (red). λ (b) (a) (d) (c) Figure 39-18 (a) An energy-level diagram for the hydrogen atom. Some of the transitions for (b) the Lyman series, (c) the Balmer series, and (d) the Paschen series. For each, the longest four wavelengths and the series-limit wavelength are plotted on a wavelength axis. Any wavelength shorter than the series-limit wavelength is allowed.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
That is, the hydrogen atom has been ionized, meaning that the electron has been removed to a distance so great that the Coulomb force on it from the nucleus is negligible. The atom can be ionized if it absorbs any wave- length shorter than the series limit. The free electron then has only kinetic energy K (= 1 2 mv 2 , assuming a nonrelativistic situation). Quantum Numbers for the Hydrogen Atom Although the energies of the hydrogen atom states can be described by the single quantum number n, the wave functions describing these states require three quantum numbers, corresponding to the three dimensions in which the electron 1206 CHAPTER 39 MORE ABOUT MATTER WAVES 1207 39-5 THE HYDROGEN ATOM A Nonquantized Nonquantized Nonquantized Nonquantized 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 2 3 4 5 6 n ∞ Energy (eV) 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 1 1 2 3 4 5 6 n ∞ Energy (eV) Lyman series Series limit 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 2 3 4 5 6 n ∞ Energy (eV) Balmer series 1 0 –2.0 –4.0 –6.0 –8.0 –10.0 –12.0 –14.0 Energy (eV) 1 λ λ λ Paschen series Series limit 2 3 4 5 6 n ∞ Series limit These are the lowest six allowed energies of the hydrogen atom. The Balmer series of wavelengths are jumps up from n = 2 (absorption) or down to n = 2 (emission). The Paschen series of wavelengths are jumps up from n = 3 (absorption) or down to n = 3 (emission). The Lyman series of wavelengths are jumps up from n = 1 (absorption) or down to n = 1 (emission). This is the shortest Balmer (series limit). λ This is the longest Balmer (red). λ (b) (a) (d) (c) Figure 39-18 (a) An energy-level diagram for the hydrogen atom. Some of the transitions for (b) the Lyman series, (c) the Balmer series, and (d) the Paschen series. For each, the longest four wavelengths and the series-limit wavelength are plotted on a wavelength axis. Any wavelength shorter than the series-limit wavelength is allowed.
eBook - ePub- Michael D. Fayer(Author)
- 2010(Publication Date)
- AMACOM(Publisher)
We used a very simple but correct mathematical method for obtaining the energy levels of the particle in the box and the wavefunctions, but the method we used is not general. For example, it cannot be used to find the energy levels and the wavefunctions for the hydrogen atom. In fact, the language we have been using, that is, wavefunctions and probability amplitude waves, comes from Schrödinger’s formulation of quantum theory. In 1925 Schrödinger presented what has come to be known as the Schrödinger Equation. The Schrödinger Equation is a complicated differential equation in three dimensions. We will not do the mathematics necessary to solve the Schrödinger equation for the hydrogen atom or other atoms or molecules. However, we will use many of the results to learn about atoms and molecules, beginning with the hydrogen atom.The solution of the hydrogen atom problem using the Schrödinger Equation is particularly important because it can be solved exactly. The hydrogen atom is a “two-body” problem. There are only two interacting particles, the proton and the electron. The next simplest atom is the helium atom, which has a nucleus with a charge of +2 and two negatively charged electrons. This is a three-body problem that cannot be solved exactly. In classical mechanics, it is also not possible to solve a three-body problem. The problem of determining the orbits of the Earth orbiting the Sun with the Moon orbiting the Earth cannot be solved exactly with classical mechanics. However, in both quantum mechanics and classical mechanics, there are very sophisticated approximate methods that permit very accurate solutions to problems that cannot be solved exactly. The fact that a method is approximate does not mean it is inaccurate. Nonetheless, because the hydrogen atom can be solved exactly with quantum theory, it provides an important starting point for understanding more complicated atoms and molecules.WHAT THE SCHRÖDINGER EQUATION TELLS US ABOUT HYDROGEN
What does the solution to the Schrödinger Equation for the hydrogen atom give? It gives the energy levels of the hydrogen atom, and it gives the wavefunctions associated with each state of the hydrogen atom. The wavefunctions are the three-dimensional probability amplitude waves that describe the regions of space where the electron is likely to be found. Schrödinger’s solution to the hydrogen atom problem gives energy levels consistent with the empirically obtained Rydberg formula. The energy levels arewhere n is the principal quantum number. It is an integer that can take on values ≥1, that is, greater than or equal to 1. The difference in energy between any two energy levels is the Rydberg formula. However, in the Schrödinger solution, RH
eBook - PDF- Gary N. Felder, Kenny M. Felder(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
For instance, the energy 320 7 The Hydrogen Atom eigenstates of a particle in an infinite square well are sines. So, you may wonder, why do we keep talking about the hydrogen atom’s eigenstates without actually writing them down? Well, since you asked . . . ψ nlm l (r,θ,φ) = m e e 2 2π 0 ¯ h 2 n 3 (n − l − 1)! 2n(n + l)! e −m e e 2 r/(4π 0 ¯ h 2 n) × m e e 2 r 2π 0 ¯ h 2 n l L 2l+1 n−l−1 m e e 2 r 2π 0 ¯ h 2 n Y m l l (θ,φ). Don’t panic! In Section 7.3 we will introduce those possibly unfamiliar independent variables, and in Section 7.4 we will start to break that formula down. We will see that some factors have important implications for the measurable behaviors of the hydrogen atom, and other factors can be ignored for our purposes. We will also outline the process of deriving that formula by solving Schrödinger’s equation. Corrections and Generalizations The quantum mechanical description of the hydrogen atom has been wildly successful. Recall that Bohr’s model was able to predict the spectrum of hydrogen by listing the energy levels, but Bohr’s quantization had no theoretical basis. Schrödinger’s equation enables us to derive those energies E n and thus correctly predict the spectrum. Quantum mechanics also allows us to derive many properties of hydrogen that the Bohr model can’t explain, such as the splitting of spectral lines (Section 7.7) and the lifetimes of excited states (found in a more advanced quantum course). Nonetheless, we should note a couple of limitations to the model as we’ve discussed it so far. The first is simple and easy to correct. You may recall from Section 4.1 that Bohr brought his original model closer in line with experiment by replacing the electron mass with the “reduced mass,” μ = m e M m e + M . (7.1) This idea comes from classical mechanics, and takes into account that the electron (mass m e ) and the nucleus (mass M) are both rotating around their common center of mass.
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