3 The Schrödinger equation

If the elec­tron in an atom of hydrogen is a standing wave, as de Broglie had assumed, why should it be con­fined to a circle? After the insight that par­ti­cles can behave like waves, which came ten years after Bohr’s quan­ti­za­tion pos­tu­late, it took less than three years for the full-​​fledged (albeit still non-​​relativistic) quantum theory to be for­mu­lated, not once but twice in dif­ferent math­e­mat­ical attire, by Werner Heisen­berg in 1925 and by Erwin Schrödinger in 1926.

Let’s take a look at where the Schrödinger equa­tion, the cen­ter­piece of non-​​relativistic quantum mechanics, comes from. Figure 2.3.1 illus­trates the prop­er­ties of a trav­eling wave ψ of Ampli­tude A and phase φ = kx − t. The wavenumber k is defined as 2π/λ; the angular fre­quency ω is given by 2π/T. Hence we can also write

φ = 2π [(xλ) − (t/​T)].

Keeping t con­stant, we see that a full cycle (2π, cor­re­sponding to 360°) is com­pleted if x increases from 0 to the wave­length λ. Keeping x con­stant, we see that a full cycle is com­pleted if t increases from 0 to the period T. (The reason why 2π cor­re­sponds to 360° is that it is the cir­cum­fer­ence of a circle or unit radius.)

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wave properties

Figure 2.3.1 The slanted lines rep­re­sent the alter­nating crests and troughs of ψ. The passing of time is indi­cated by the upward-​​moving dotted line, which rep­re­sents the tem­poral present. It is readily seen that the crests and troughs move toward the right. By focusing on a fixed time, one can see that a cycle (crest to crest, say) com­pletes after a dis­tance λ. By focusing on a fixed place, one can see that a cycle com­pletes after a time T.

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The math­e­mat­i­cally sim­plest and most ele­gant way to describe ψ is to write

ψ = [A:φ] = [A:kx − ωt].

This is a com­plex number of mag­ni­tude A and phase φ. It is also a func­tion ψ(x,t) of one spa­tial dimen­sion (x) and time t.

We now intro­duce the oper­a­tors ∂x and ∂t. While a func­tion is a machine that accepts a number (or sev­eral num­bers) and returns a (gen­er­ally dif­ferent) number (or set of num­bers), an oper­ator is a machine that accepts a func­tion and returns a (gen­er­ally dif­ferent) func­tion. All we need to know about these oper­a­tors at this point is that if we insert ψ into ∂x, out pops ikψ, and if we insert ψ into ∂t, out pops −iωψ:

xψ = ikψ,     ∂tψ = −iωψ.

If we feed ikψ back into ∂x, out pops (not unex­pect­edly) (ik)2ψ = −k2ψ. Thus

(∂x)2ψ = −k2ψ.

Using Planck’s rela­tion E = ω and de Broglie’s rela­tion p = h/​λ = k to replace ω and k by E and p, we obtain

tψ = −i(E/)ψ,     ∂xψ = i(p/)ψ,   (∂x)2ψ = −(p/​)2ψ,

or

(2.3.1)   Eψ = itψ,     pψ = (/​i)∂xψ,   p2ψ = −2(∂x)2ψ.

We now invoke the clas­sical, non-​​relativistic rela­tion between the energy E and the momentum p of a freely moving particle,

(2.3.2)   E = p2/​2m,

where m is the particle’s mass. We shall dis­cover the origin of this rela­tion when taking on the rel­a­tivistic theory. The right-​​hand side is the particle’s kinetic energy.

Mul­ti­plying Eq. (2.3.2) by ψ and using Eqs. (2.3.1), we get

(2.3.3)   itψ = −(2/​2m) (∂x)2ψ.

This is the Schrödinger equa­tion for a freely moving par­ticle with one degree of freedom — a par­ticle capable of moving freely up and down the x-​​axis. We shouldn’t be sur­prised to find that Eq. (2.3.3) imposes the fol­lowing con­straint on ψ:

(2.3.4)   ω = k2/​2m.

This is nothing else than Eq. (2.3.2) with E and p replaced by ω and k according to the rela­tions of Planck and de Broglie.

We have started with a spe­cific wave func­tion ψ. What does the gen­eral solu­tion of Eq. (2.3.3) look like? The ques­tion is readily answered by taking the fol­lowing into account: If ψ1 and ψ2 are solu­tions of Eq. (2.3.3), then for any pair of com­plex num­bers a,b the func­tion ψ = aψ1 + bψ2 is another solu­tion. The gen­eral solu­tion, accord­ingly, is

(2.3.5)   ψ(x,t) = (1/√(2π)) ∫dk [a(k):kx − ω(k)t].

The factor (1/√(2π)) ensures that the prob­a­bil­i­ties cal­cu­lated with the help of ψ are nor­mal­ized (that is, the prob­a­bil­i­ties of all pos­sible out­comes of any given mea­sure­ment add up to 1). The symbol ∫dk indi­cates a sum­ma­tion over all values of k from k=−∞ to k=+∞: every value con­tributes a com­plex number a(k)[1:kx − ω(k)t], where ω(k) is given by Eq. (2.3.4).

If the par­ticle is moving under the influ­ence of a poten­tial V, the poten­tial energy qV (q being the particle’s charge) needs to be added to the kinetic energy (the right-​​hand side of Eq. 2.3.2). The Schrödinger equa­tion then takes the form

(2.3.6)   itψ = −(2/​2m) (∂x)2ψ + qVψ.

Its gen­er­al­iza­tion to three-​​dimensional space is now straightforward:

(2.3.7)   itψ = −(2/​2m) [(∂x)2 + (∂y)2 + (∂z)2]ψ + qVψ.

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