6 The hydrogen atom

Using the time-​​independent Schrödinger equa­tion with the poten­tial energy term V = –e2/​r, where e is the absolute value of the charge both of the elec­tron and of the proton, we again find that bound states exist only for spe­cific values of the total energy E. These are exactly the values that Bohr had obtained via his 1913 pos­tu­late.

Just as fac­tor­izing ψ(x,y,z,t) into Ψ(x,y,z) and [1:−Et/ℏ] led to a time-​​independent Schrödinger equa­tion and a dis­crete set of values En, so fac­tor­izing Ψ(r,φ,θ) — which is Ψ(x,y,z) in polar coor­di­nates — into ψ(r,θ) and [1;Lzφ/​ℏ] leads to a φ-inde­pen­dent Schrödinger equa­tion and a dis­crete set of values Lz.


polar coordinates

Figure 2.6.1 Polar coordinates


The φ-inde­pen­dent Schrödinger equa­tion con­tains a real para­meter whose pos­sible values are given by L(L+1)ℏ2, where L is an integer sat­is­fying the con­di­tion 0 ≤ L ≤ n-​​1. The pos­sible values of Lz are inte­gers sat­is­fying the inequality |Lz| ≤ L. The pos­sible com­bi­na­tions of the quantum num­bers n, L, and Lz are thus

n = 1xxxL = 0xxxLz = 0

n = 2xxxL = 0xxxLz = 0

n = 2xxxL = 1xxxLz = –1, 0, +1

n = 3xxxL = 0xxxLz = 0

n = 3xxxL = 1xxxLz = –1, 0, +1

n = 3xxxL = 2xxxLz = –2, –1, 0, +1, +2


All of these states are sta­tionary. n is known as the prin­cipal quantum number, L as the angular momentum (or orbital, or azimuthal) quantum number, and Lz as the mag­netic quantum number (hence the letter m is often used instead).

States with L = 0, 1, 2, 3 were orig­i­nally labeled s, p, d, f — for “sharp,” “prin­cipal,” “dif­fuse,” and “fun­da­mental,” respec­tively. The pur­pose of these let­ters was to char­ac­terize spec­tral lines. States with higher L follow the alphabet (g, h, …). Figure 2.6.2 maps the radial depen­den­cies of the first three s states, which are spher­i­cally sym­metric. The plots can be iden­ti­fied by the number N of their nodes (N = n-​​1).


s orbitals

Figure 2.6.2 Radial depen­den­cies of the states with quantum num­bers 1s, 2s, and 3s.


Fig­ures 2.6.3 and 2.6.4 plot the posi­tion prob­a­bility dis­tri­b­u­tions defined by some non-​​spherical sta­tionary states with m = 0. Figure 2.6.3 empha­sizes the fuzzi­ness of these orbitals at the expense of their rota­tional sym­metry. By plot­ting sur­faces of con­stant prob­a­bility, Figure 2.6.4 empha­sizes their 3-​​dimensional shape at the expense of their fuzziness.


hydrogen orbitals ray-traced

Figure 2.6.3 The posi­tion prob­a­bility dis­tri­b­u­tions asso­ci­ated with the fol­lowing orbitals. First row: 2p0, 3p0, 3d0. Second row: 4p0, 4d0, 4f0. Third row: 5d0, 5f0, 5g0. Imag­ining method: ray-​​traced. Not to scale.


hydrogen orbitals constant-probability surfaces

Figure 2.6.4 The posi­tion prob­a­bility dis­tri­b­u­tions asso­ci­ated with the same orbitals as in Figure 2.6.3. Imag­ining method: sur­face of con­stant prob­a­bility. Not to scale.


It must be stressed that what we see in these images is nei­ther the nucleus nor the elec­tron but the fuzzy posi­tion of the elec­tron rel­a­tive to the nucleus. Nor do we see this fuzzy posi­tion “as it is.” What we see is the plot of a posi­tion prob­a­bility dis­tri­b­u­tion. This is defined by out­comes of three mea­sure­ments, deter­mining the values of n, L, and Lz, and it defines a fuzzy posi­tion by deter­mining the prob­a­bil­i­ties of the pos­sible out­comes of a sub­se­quent mea­sure­ment of the posi­tion of the elec­tron rel­a­tive to the nucleus.

Here is how such a prob­a­bility can be cal­cu­lated. Imagine a small region V of space in the vicinity of the nucleus — so small that the prob­a­bility den­sity ρ (prob­a­bility per unit volume) inside it can be con­sid­ered con­stant. The prob­a­bility of finding the elec­tron inside V (if the appro­priate mea­sure­ment is made) is the product ρV. If the gray inside V is a lighter shade, this prob­a­bility is lower; if it’s a darker shade, this prob­a­bility is higher. To cal­cu­late the prob­a­bility asso­ci­ated with a larger region, divide it into suf­fi­ciently many suf­fi­ciently small regions and add up the prob­a­bil­i­ties asso­ci­ated with them.

Since the depen­dence on φ is con­tained in the factor [1;Lzφ/​ℏ], it cannot be seen in plots of |Ψ(r,φ,θ)|2. To make this depen­dence vis­ible, it is cus­tomary to replace the com­plex number [1;Lzφ/​ℏ] by its real part, as has been done in Figure 2.6.5.


Orbitals with non-zero m

Figure 2.6.5 Orbitals with non-​​zero m. First row: 4f1, 5f1. Second row: 5f2, 5f3. Third row: 5g1, 5g3. Fourth row: 5g3, 5g4. Not to scale.