24

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by the Schrödinger equation $$i\hbar\frac{\partial}{\partial t}|\psi_t\rangle = H|\psi_t\rangle$$ where $H$ is the Hamilton operator (for the free particle we have $H=-\frac{\hbar^2}{2m}\Delta$).

Now I have often seen used spaces like $\mathcal{H}=L^2(\mathbb{R}^3)$ (in the case of a single particle), but I was wondering whether this is correct or not. In fact shouldn't we require to be able to derivate $\left|\psi_t\right>$ twice in $x$ and thus choose something like $\mathcal{H} = H^2(\mathbb{R}^3)$?

If we treat directly $\psi(t,x) := \psi_t(x)$, shouldn't we require them to be in something like $H^1(\mathbb{R};H^2(\mathbb{R}^3))$? i.e., functions in $H^1(\mathbb{R})$ with values in $H^2(\mathbb{R}^3)$, e.g. the function $t\mapsto\psi_t$.

Shuhao Cao
  • 19,689
Daniel Robert-Nicoud
  • 30,663
  • Quantum mechanics rests on the basic fact that wave functions are square integrable and the conservation of probability. My question is, are the spaces you mentioned subspaces of the Hilbert space? – Torsten Hĕrculĕ Cärlemän Jul 26 '13 at 20:51
  • 1
    @TorstenHĕrculĕCärlemän The space $L^2(X)$ is the space of square integrable functions on $X$, and you can look at the space $H^k(X)$ as the space of all functions on $X$ which are square integrable and such that all their derivatives up to order $k$ are also square integrable. Look up Sobolev spaces if you want more informations on the subject. – Daniel Robert-Nicoud Jul 26 '13 at 20:57
  • One more question, are functions in $L^2(X)$ not twice differentiable as in the Sobolev space $H^2(X)$? – Torsten Hĕrculĕ Cärlemän Jul 26 '13 at 21:07
  • @TorstenHĕrculĕCärlemän No. The only requirement you have on functions of $L^2(X)$ is to be square integrable. You could for example take $X = [0,1]$ and $f(x) = \chi_{\mathbb{Q}\cap[0,1]}(x)$ (the characteristic function of $\mathbb{Q}\cap[0,1]$) would be in $L^2([0,1])$. However $f$ is nowhere differentiable. – Daniel Robert-Nicoud Jul 26 '13 at 21:10
  • @TorstenHĕrculĕCärlemän A note: I've cheated a bit with my previous example. In fact the elements of $L^2(X)$ are equivalence classes of functions, where two functions are equivalent if their difference squared integrates to zero. In fact you can always find a representative which is at least piecewise continuous. But I assure you there are functions in $L^2$ which are nowhere differentiable. – Daniel Robert-Nicoud Jul 26 '13 at 21:15
  • I have them in one of my own answers. Lurking in suburban Hilbert space :D. Thanks for your patience, cher ami! To work. – Torsten Hĕrculĕ Cärlemän Jul 26 '13 at 21:29
  • Dear Daniel, As a bit of a side remark, while it is interesting to think about mathematically rigorous formulations of quantum mechanics (and other physical theories), this is a bit orthogonal to the way physicists work and think (in my experience). E.g. a particle with no forces acting is typically modelled as a plane wave $e^{i k x}$, which is not in $L^2$. Its Fourier transform is a delta function supported at $k$ (so the momentum of the particle being modelled is fixed to be $k$); a delta function is not a function at all in the rigorous sense, but a distribution. In short, I think ... – Matt E Jul 26 '13 at 22:29
  • ... that in physics it is convenient to be vague, and flexible, about the space being worked in, and to be willing to modify it as necessary for analyzing a particular situation. So pinning things down precisely mathematically can be of limited use in understanding in what physicists are doing (although certainly it can be helpful to those of us with a mathematical bent!). Regards, – Matt E Jul 26 '13 at 22:31
  • P.S. If you have never looked at von Neumann's book on quantum mechanics, you might find it interesting. My memory is that he does address exactly the kind of question you are asking, and in particular develops the spectral theory of unbounded operators as a way of thinking about the Schrodinger equation as an operator defined on $L^2$. (One aspect of the theory of unbounded operators is that the domain may not be the entire Hilbert space.) On the other hand, my sense is that in modern analysis the theory of unbounded operators doesn't occur as much; instead people change the Hilbert ... – Matt E Jul 26 '13 at 22:34
  • ... space so that the operator becomes bounded (e.g. replacing $L^2$ by a Sobolev space). – Matt E Jul 26 '13 at 22:34
  • @Matt You can define weighted $L^2$ spaces that contain plane waves. The only reference that comes to my mind though is the book by Leis on Wave equations – Guido Kanschat Jul 27 '13 at 10:48

4 Answers4

25

Unfortunately, taking $\mathcal{H}=H^2(\mathbb{R}^3)$ won't work. If $f\in H^2(\mathbb{R}^3)$ then, in general, it is not true that $\Delta f\in H^2(\mathbb{R}^3)$. This is a problem, because the Hamiltonian must be an operator from $\mathcal{H}$ to $\mathcal{H}$.

The only mathematically sound way out is taking $\mathcal{H}=L^2(\mathbb{R}^3)$ and considering the Laplacian as an unbounded operator, meaning that it is not defined on the whole $L^2(\mathbb{R}^3)$ space but only on a dense subspace. This idea dates back to von Neumann.

Giuseppe Negro
  • 34,508
  • Nice answer, uses much higher tech than mine, and addresses the point more directly, I was gonna say something about the function space involved in the eigenvalue problem $L u = \lambda u $ is pretty different from the elliptic problem $Lu = f$, then lost it in writing the examples. +1 – Shuhao Cao Jul 26 '13 at 23:50
7

For the linear Schrödinger equation on $\mathbb{R}^n$ $$ \begin{cases} \partial_t u = i\Delta u \\ u(x,0)=u_0(x) \end{cases} $$ the solution propagator $e^{it\Delta}u_0=(e^{-4\pi it|\xi|^2}\hat{u_0})^{\vee}$ forms a unitary group on $H^s$ for every $s \in \mathbb{R}$ as it's a Fourier multiplier of modulus 1. Thus we wouldn't have a solution in $H^1(\mathbb{R}:H^s(\mathbb{R}^n))$ (no decay in time of $H^s$-norm), but $C(\mathbb{R}:H^s(\mathbb{R}^n))$.

For conjugate exponents $p,p'$ with $p' \in [1,2]$ we have that the propagator $e^{it\Delta} : L^{p'}(\mathbb{R}^n) \rightarrow L^p(\mathbb{R}^n)$ is continuous with $$ \|e^{it\Delta}f\|_p \leq c|t|^{-n/2(1/p'-1/p)}\|f\|_{p'}. $$ This can be proved by interpolating the $L^2$ isometry result along with the case $p=1,p'=\infty$ (which is handled by Young's inequality). This leads to the so called global smoothing effects or Strichartz estimates. For example, the above result along with Hardy-Littlewood-Sobolev inequality implies $$ \left(\int_\mathbb{R} \|e^{it\Delta}f\|_p^q \; dt \right)^{1/q} \leq c\|f\|_2 $$ given some relations on $n,p$ and $q$.

Now you can attack nonlinear problems like the semilinear Schrödinger equation $$ \begin{cases} i\partial_t u = -\Delta u - \lambda |u|^{\alpha-1}u \\ u(x,0)=u_0(x) \end{cases} $$ for $\alpha>1$. One approach is to use Duhamel's principle and solve the (weak) integral formulation $$ u(t) = e^{it\Delta}u_0 + i\lambda \int_0^t e^{i(t-t')\Delta}(|u|^{\alpha-1}u)(t') \; dt' $$ via the contraction mapping principle on an appropriate space. The Strichartz estimates play a key role in deriving the contraction property. For a result about local existence for $H^1$ data, see this paper of Kato in which he produces a solution in $C(I:H^1) \cap C^1(I:H^{-1})$ for some interval $I$ given assumptions on the nonlinear term. More references may be found on the Dispersive Wiki. For more mathematical details on Strichartz estimates and the contraction argument, check out Cazenave's Semilinear Schrödinger Equations.

dls
  • 4,776
  • This answer is meant to provide mathematical background to results you may encounter in physics literature. – dls Jul 26 '13 at 23:19
5

My first attempt at a weak formulation for your problem would be: \begin{equation} \left(i\partial_t \psi,\phi\right) - \tfrac{\hbar}{2m}\left(\nabla \psi,\nabla\phi\right)=0, \end{equation} where $(.,.)$ is the $L^2$-inner product. This would (analogous to parabolic equations) require a space like $$ W = \left\{v\in L^2(I;V)\Big| i\partial_t v \in L^2(I;V')\right\} $$ Here, $I$ is a suitable time interval, $V=H^1_0(R^3)$, and $V'$ its dual.

Guido Kanschat
  • 336
  • Yes, your space $V$ is basically what I mean by $H^1(\mathbb{R};H^2(\mathbb{R}^3))$. Notice that you don't need $i\partial_tv\in L^2(I,V')$ (in the sense that you don't need the dual $V'$) because the space is Hilbert. – Daniel Robert-Nicoud Jul 26 '13 at 20:53
  • The space $W$ does not involve second derivatives, since due to integration by parts, only a single derivative is needed – Guido Kanschat Jul 26 '13 at 21:01
  • Yes, if you look for weak solutions. Also, shouldn't those be in fact $H^1$ instead of $L^2$ in your definition of $W$? – Daniel Robert-Nicoud Jul 26 '13 at 21:03
  • The distinction of $V$ and $V'$ is important in the definiton of $W$. These two spaces are isomorphic by the Riesz representation theorem, but not equal, see [http://math.stackexchange.com/a/26291/87363]. – Guido Kanschat Jul 26 '13 at 21:34
  • Ok, yeah, you're right. I just never got the hang of the difference... Anyway, this makes sense from the point of view of PDEs, but do you know by any chance some references or similar stuff treating QM in full formality with all the correct spaces? I would be interested to see how it is really done. – Daniel Robert-Nicoud Jul 26 '13 at 21:38
4

Short answer: it is technically okay because smooth functions are just subset of the $L^2(\mathbb{R}^3)$.

Long answer: $L^2(\mathbb{R}^3)$ addresses more of the inner product than the differentiability: for normally we want the our eigenfunctions $\{|\psi_j \rangle \}$ are orthogonal to each other, also normalized spacially in $L^2(\mathbb{R}^3)$. $$ \langle \psi_i |\psi_j\rangle = \int_{\mathbb{R}^3} \overline{\psi_i}(t,x)\psi_j(t,x) \,dx = 0, \quad i\neq j. $$ Also normalized: $$ \langle \psi_i |\psi_i\rangle = \int_{\mathbb{R}^3} \overline{\psi_i}(t,x)\psi_i(t,x) \,dx = 1. $$ Such that $$ \frac{d}{dt}\langle \psi_i |\psi_i\rangle = 0 $$ If eigenfunctions satisfy above property, then for any normalized wave function $|\psi\rangle$ spanned using inner product has a diagonal form: $$ |\psi\rangle = \sum^\infty_{i=1} \langle \psi_i |\psi \rangle\, |\psi_i \rangle, $$ translated to mathematical notation: $$ \psi(t,x) = \sum^\infty_{i=1} \alpha_i(t) \,\psi_i(t,x), $$ where $$ \alpha_i(t) = \int_{\mathbb{R}^3} \overline{\psi_i}(t,x)\psi (t,x)dx, $$ so that we can have a physical interpretation, namely, the probability amplitude of the system described by wave function $\psi(t,x)$ being at that state $i$ is $\alpha_i$, a.k.a. $|\alpha_i|^2$ is the probability the system at state $i$ at time $t$. All these benefit from the orthonormality of the basis.

The $\psi_i$'s are found by solving the eigenvalue problem: $$ H\psi_i = E_i \psi_i, $$ where $E_i$ is the eigenvalue for the operator $H$. Mathematical analogy would be the eigenfunctions of minus Laplacian on a square: $$ -\Delta u = k^2u, $$ the eigenfunctions are orthonormal in $L^2$, and orthorgonal in $H^1$.

Differentiability is not the main concern in the quantum world AFAIK.

Firstly, for Bochner space you mentioned, the wave function is normally assumed to be smoothly evolving over time, the obtaining of the wave function for simple models normally follows some Ansatzes, plane wave, decay, etc. The functions we obtain from these Ansatzes are always smooth.

For example, think 1D single particle in a box model: $$ i\hbar\frac{\partial}{\partial t} \psi(t,x) = H\left|\psi \right> = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \psi(t,x) \tag{$\star$} $$

there is a particle at some state $E$ in $x\in [-1,0]$, then at some time $t_0$ the box length is doubled, $x\in [-1,1]$. What we do is to expand the wave function $\psi(t_0,x)$ at $t_0$ in the old box using the new eigenfunctions $\phi_i$ in the new box $[-1,1]$: $$\psi(t_0,x) = \sum_{i=1}^{\infty} \langle\phi_i(x)|\psi(t_0,x)\rangle \phi_i(x),\tag{1}$$ where the new eigenfunctions $\phi_i$ solves the eigenvalue problem for $(\star)$ in the new box $$ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\phi_i(t,x) = E_i \phi_i(t,x) , \quad \text{ for }x \in[-1,1]. $$ Then $$ \psi(t_0+\Delta t,x) = \sum_{i=1}^{\infty} \langle\phi_i(x)|\psi(t_0,x)\rangle \phi_i(x)e^{ -iE_n \Delta t/\hbar } \tag{2} $$ The question is: Does the wave function evolve smoothly? The answer is yes, well, methodologically, there is no non-smoothness existing. For (1) can be expanded in old eigenfunctions or in new eigenfunctions. An exercise for you is to check what happened when $\Delta t\to 0$ in (2).

Second, the differentiability in space is normally infinite within the domain of interest, and continuous up to the boundary of the domain of interest, zero outside. Just think of that particle in the box model (infinite potential wall): the eigenfunction is set to be zero on boundaries $x=1,-1$.

Shuhao Cao
  • 19,689