Stationary States of the Hydrogen Atom

In his 1913 papers Bohr developed a theory of the stationary states of the hydrogen atom. According to his theory the electron was to be considered as moving in a circular orbit about the proton. The amount of angular momentum for a stationary tate was assumed by Bohr to be equal to nh/2w, with n = 1, 2, 3, . J n Appendix II there is given the derivation of the energy values fcr the Bohr circular orbits. For... 

In this chapter we discuss angular momentum, and in the next chapter we show that for the stationary states of the hydrogen atom the magnitude of the electron s angular momentum is constant. As a preliminary, we consider what criterion we can use to decide which properties of a system can be simultaneously assigned definite values. 

Bohr also discovered a method of calculating the energy of the stationary states of the hydrogen atom, with use of Planck s constant. He... 

If the zero of the electronic energy scale is taken as the state in which the electron is at an infinite distance from the nucleus, all eigenvalues for the stationary states of the hydrogen atom correspond to stabilizing energies ... 

Bohr theory, the radius of the circular orbit of the electron in the ground state of the hydrogen atom (Z = 1) with a stationary nucleus. Except in Section 6.5, where this substitution is not appropriate, we replace fx by and by ao in the remainder of this book. 

In most applications, the reduced mass is sufficiently close in value to the electronic mass me that it is customary to replace pt in the expressions for the energy levels and wave functions by me. The parameter aM = h2/fie 2 is thereby replaced by ao = h2/mee 2. The quantity ao is, according to the earlier Bohr theory, the radius of the circular orbit of the electron in the ground state of the hydrogen atom (Z = 1) with a stationary nucleus. Except in Section 6.5, where this substitution is not appropriate, we replace ft by me and atl by ao in the remainder of this book. 

From his relation (4 ) De Broglie was able immediately to draw a very important conclusion regarding the stationary states of a hydrogen atom. His result agreed with that of Bohr. Whereas however in Bohr s theory the stationary states were introduced completely ad hoc to explain the spectra, they follow with De Broglie logically and obviously from the basic hypothesis. 

A stationary-state wave function is an eigenfunction of the Hamiltonian operator H = T + V. Students sometimes erroneously believe that ip is an eigenfunction of T and of V. For the ground state of the hydrogen atom, verify directly that tp is not an eigenfunction of T or of V, but is an eigenfunction of f -I- V. Can you think of a problem we solved where (/> is an... 

Explain verbally why (L ) and (L ) are independent of Z for all stationary states of the hydrogen-like atom. Since the average distance from the nucleus depends on Z, what does this mean about the average speed of the electron around the nucleus for the 211 state ... 

Soon after Bohr developed his initial configuration Arnold Sommerfeld in Munich realized the need to characterize the stationary states of the electron in the hydrogen atom by. means of a second quantum number—the so-called angular-momentum quantum number, Bohr immediately applied this discovery to many-electron atoms and in 1922 produced a set of more detailed electronic configurations. In turn, Sommerfeld went on to discover the third or inner, quantum number, thus enabling the British physicist Edmund Stoner to come up with an even more refined set of electronic configurations in 1924. 

Solution of (12) gives the complete non-relativistic quantum-mechanical description of the hydrogen atom in its stationary states. The wave function is interpreted in terms of... 

The theory discussed here gives a special role to the stationary states of the molecular hamiltonian. In particular, there are stationary electronic states, not a set of electrons. For example, the hydrogen atom cannot be seen as formed by one proton plus one electron. It is the electronic spectra which define it, not the model we use to calculate the energy levels and wave functions. This may sound strange but consider a thermal neutron. This system decomposes into one proton plus an electron and a neutrino. One cannot say that a neutron is made of such particles. Matter may exist in different kinds of stationary states processes can be seen as changes among them. 

The incompatibility of Rutherford s planetary model, based soundly on experimental data, with the principles of classical physics was the most fundamental of the conceptual challenges facing physicists in the early 1900s. The Bohr model was a temporary fix, sufficient for the interpretation of hydrogen (H) atomic spectra as arising from transitions between stationary states of the atom. The stability of atoms and molecules finally could be explained only after quantum mechanics had been developed. 

Figure 7.9 Quantum staircase. In this analogy for the energy levels of the hydrogen atom, an electron can absorb a photon and jump up to a higher step (stationary state) or emit a photon and jump down to a lower one. But the electron cannot lie between two steps.

We proceed now to those spectra not of the hydrogen type. As wc have already mentioned in 21 we endeavour, following Bohr, to ascribe the production of these spectra to transitions between stationary states of the atom, each of these stationary states being characterised essentially by the motion of a single radiating or series electron in an orbit under the influence of the core, which is represented approximately by a central field of force. This conception explains some of the most important regularities of the series of spectra, namely, the existence of several series, each of which is more or less similar to the hydrogen type, and the possibility of combinations between these. 

Until now in our quantum mechanical discussions we have described the stationary or time-independent states of a system. Furthermore, our language was such as to imply that we had sufficient information about the system to know that it was in a particular state described by a particular set of quantum numbers. For example, in the case of the harmonic oscillator we spoke as though we knew that the oscillator was in the pth state with wave function i// , and energy = (v A- j)hv or, in the case of the hydrogen atom, that it was in a state described by the set of numbers n, /, m. This approach is very useful in a first discussion of quantum mechanical properties of various kinds of systems. However, we do not have reason to presuppose that a system is in a particular quantum state. 

The quantum-mechanical treatment of the hydrogen atom has been thoroughly worked out. A number of stationary (non-time variable) states are possible. Each state may be... 

As many authors note, the quantization of angular momentum assumed by Bohr as weU as the notion that electrons in stationary states do not radiate was somewhat ad hoc and only justified later by Erwin Schrodinger s approach to calculating the energy of the hydrogen atom. 

A rough understanding of the spectrum and structure of atoms was first achieved by Niels Bohr in Copenhagen. He realized that a classical treatment of the hydrogen atom, similar to planet motion, would not give rise to a discrete spectrum, not even a stable atom. Therefore, he introduced stationary states for the first time. This was a hint of what was to come in the form of quantum mechanics about 10 years later. His principal assumptions are worth citing, since the main differences between classical theory and what is reqnired by a novel theory are clearly stated ... 

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