16b. Schr\"odinger equation

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Time now to get into quantum mechanics. This is a bit of a nightmare, and I am afraid that I will have to ask again the cat. With him, things going better since last time, we sort of agree now on multivariable calculus, following a long philosophical debate, and my willingness to start sharing his diet, namely raw mice and birds. But let's see now in relation with this highly divisive topic which is quantum mechanics: \begin{cat} Schrödinger. \end{cat} Thanks cat, this is actually quite surprising, and I am particularly pleased to hear this. So, before getting back to what Heisenberg was saying, based on Lyman, Balmer, Paschen, namely developing some sort of “matrix mechanics”, let us hear as well the point of view of Schrödinger, which came a few years later. His idea was to forget about exact things, and try to investigate the hydrogen atom statistically. Let us start with: \begin{question} In the context of the hydrogen atom, assuming that the proton is fixed, what is the probability density [math]\varphi_t(x)[/math] of the position of the electron [math]e[/math], at time [math]t[/math],

[[math]] P_t(e\in V)=\int_V\varphi_t(x)dx [[/math]]

as function of an intial probability density [math]\varphi_0(x)[/math]? Moreover, can the corresponding equation be solved, and will this prove the Bohr claims for hydrogen, statistically? \end{question} In order to get familiar with this question, let us first look at examples coming from classical mechanics. In the context of a particle whose position at time [math]t[/math] is given by [math]x_0+\gamma(t)[/math], the evolution of the probability density will be given by:

[[math]] \varphi_t(x)=\varphi_0(x)+\gamma(t) [[/math]]

However, such examples are somewhat trivial, of course not in relation with the computation of [math]\gamma[/math], usually a difficult question, but in relation with our questions, and do not apply to the electron. The point indeed is that, in what regards the electron, we have: \begin{fact} In respect with various simple interference experiments:

• The electron is definitely not a particle in the usual sense.
• But in most situations it behaves exactly like a wave.
• But in other situations it behaves like a particle.

\end{fact} Getting back now to the Schrödinger question, all this suggests to use, as for the waves, an amplitude function [math]\psi_t(x)\in\mathbb C[/math], related to the density [math]\varphi_t(x) \gt 0[/math] by the formula [math]\varphi_t(x)=|\psi_t(x)|^2[/math]. Not that a big deal, you would say, because the two are related by simple formulae as follows, with [math]\theta_t(x)[/math] being an arbitrary phase function:

[[math]] \varphi_t(x)=|\psi_t(x)|^2\quad,\quad \psi_t(x)=e^{i\theta_t(x)}\sqrt{\varphi_t(x)} [[/math]]

However, such manipulations can be crucial, raising for instance the possibility that the amplitude function satisfies some simple equation, while the density itself, maybe not. And this is what happens indeed. Schrödinger was led in this way to:

\begin{claim}[Schrödinger] In the context of the hydrogen atom, the amplitude function of the electron [math]\psi=\psi_t(x)[/math] is subject to the Schrödinger equation

[[math]] ih\dot{\psi}=-\frac{h^2}{2m}\Delta\psi+V\psi [[/math]]

[math]m[/math] being the mass, [math]h=h_0/2\pi[/math] the reduced Planck constant, and [math]V[/math] the Coulomb potential of the proton. The same holds for movements of the electron under any potential [math]V[/math]. \end{claim} Observe the similarity with the wave equation [math]\ddot{\varphi}=v^2\Delta\varphi[/math], and with the heat equation [math]\dot{\varphi}=\alpha\Delta\varphi[/math] too. Many things can be said here. Following now Heisenberg and Schrödinger, and then especially Dirac, who did the axiomatization work, we have:

Definition

In quantum mechanics the states of the system are vectors of a Hilbert space [math]H[/math], and the observables of the system are linear operators

[[math]] T:H\to H [[/math]]

which can be densely defined, and are taken self-adjoint, [math]T=T^*[/math]. The average value of such an observable [math]T[/math], evaluated on a state [math]\xi\in H[/math], is given by:

[[math]] \lt T \gt = \lt T\xi,\xi \gt [[/math]]

In the context of the Schrödinger mechanics of the hydrogen atom, the Hilbert space is the space [math]H=L^2(\mathbb R^3)[/math] where the wave function [math]\psi[/math] lives, and we have

[[math]] \lt T \gt =\int_{\mathbb R^3}T(\psi)\cdot\bar{\psi}\,dx [[/math]]

which is called “sandwiching” formula, with the operators

[[math]] x\quad,\quad-\frac{ih}{m}\nabla\quad,\quad-ih\nabla\quad,\quad -\frac{h^2\Delta}{2m}\quad,\quad -\frac{h^2\Delta}{2m}+V [[/math]]

representing the position, speed, momentum, kinetic energy, and total energy.

In other words, we are doing here two things. First, we are declaring by axiom that various “sandwiching” formulae found before by Heisenberg, involving the operators at the end, that we will not get into in detail here, hold true. And second, we are raising the possibility for other quantum mechanical systems, more complicated, to be described as well by the mathematics of the operators on a certain Hilbert space [math]H[/math], as above.

So, this was the story of early quantum mechanics, over-simplified as to fit here in a few pages. For more, you can check Feynman for foundations, and everything, including for some nice pictures and explanations regarding Fact 16.11. You have as well Griffiths or Weinberg , for further explanations on Definition 16.13, not to forget Dirac's original text , and all this is discussed as well in my book .

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].