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The operator $P=-i\hbar\frac{d}{dx}$, is a symmetric operator in the domain $$D(P)=\left\{f(x) \big|f\in L_2[0, a], f(0)=f(a)=0\right\}$$ i.e. the domain is the subspace of square-integrable functions in the interval $0\leq x\leq a$ that vanish at the endpoints $x=0$ and $x=a$. The adjoint turns out to be $P^\dagger=i\hbar\frac{d}{dx}$.

The operator $P$ is symmetric but not self-adjoint because $D(P^\dagger)\neq D(P)$ where $$D(P^\dagger)=\left\{g(x) \big|g\in L_2[0,a]\right\}$$

If we want to apply this idea to the problem of a quantum particle in a box (infinite square well), then the wavefunctions representing all possible states of the system are both square-integrable and vanish at endpoints. So $-i\hbar\frac{d}{dx}$ is automatically self-adjoint. We don't need a self-adjoint extension. Am I right?

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    $\begingroup$ Why do you think that $i\mathrm d/\mathrm dx$ is "automatically" self-adjoint with the domain as above, when you just have argued it is not?! $\endgroup$ Commented Mar 3, 2023 at 18:24
  • $\begingroup$ Sorry for the confusion. I am just beginning to learn these intricacies. My point is the Hilbert space for the problem of a particle in a box, contains only those functions that are both square-integrable and vanish at endpoints. Doesn't that make the domains equal? $\endgroup$ Commented Mar 3, 2023 at 18:26
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    $\begingroup$ You forgot to include the differentiability condition on the domain... $\endgroup$ Commented Mar 3, 2023 at 18:35
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/362305/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented Mar 3, 2023 at 18:45
  • $\begingroup$ @hft While I agree that the title question is answered by the post you've linked, I don't think it is an exact duplicate. Rather, the OP has an error in their line of thought. I think they are aware of the fact that the operator is not self-adjoint. So I don't see this as a duplicate here. $\endgroup$ Commented Mar 3, 2023 at 19:49

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You forgot to include the differentiability condition for the domain of $P$ and $P^\dagger$. Indeed, suppose our Hilbert space is $L^2(I)$ with $I:=[0,2\pi]$.$^\ddagger$ Define the domain of $P:=-i\frac{\mathrm d }{\mathrm dx}$ as

$$\mathcal D(P):=\{\psi\in C^1(I) |\psi(0)=0 = \psi(2\pi)\} \quad .$$ One can show that

$$\mathcal D(P^\dagger)= H^1(I) \quad , $$

where $H^1$ denotes the Sobolev space. It holds that $C^1(I)\subsetneq H^1(I)\subsetneq L^2(I)$. As such, we have $\mathcal D(P) \subsetneq \mathcal D(P^\dagger)$ and thus $P$ is not self-adjoint.


$^\ddagger$ It does not make sense to define the Hilbert space as something like $$\tilde L^2(I) := \{\psi\in L^2(I)|\psi(0)=\psi(2\pi) = 0\} \quad ,$$ because actually we are dealing here with equivalence classes of wave functions, which cannot see points of measure zero. Rather, the infinite potential at the boundaries of the box is taken into account by the domain of the Hamiltonian.

References and further reading:

  1. F.P. Schuller. Lectures on Quantum Theory. Lecture Notes, chapter 9. Here is a video of the relevant lecture: YouTube.

  2. The classic PSE thread: What's the deal with momentum in the infinite square well?

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  • $\begingroup$ Does the Hilbert space of the problem contain states of the system that are square-integrable but non-vanishing at $x=0$ and $x=a$? $\endgroup$ Commented Mar 5, 2023 at 10:10
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    $\begingroup$ @Solidification Yes. As I have tried to point out, you cannot really define the Hilbert space as "functions which vanish at the endpoints" (for various mathematical reasons) - see also the great other answer here by J. Murray for why this is not possible. But for the particle in the box problem, we define the Hamiltonian on a domain on which functions vanish at the boundaries (see e.g. reference 2 in my answer). $\endgroup$ Commented Mar 5, 2023 at 10:13
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    $\begingroup$ @Solidification Let me add: To be very precise, however, one shouldn't talk about functions as elements of the $L^2$ space- the elements are equivalence classes and thus we cannot really talk about functions which have a particular behavior at single points in general. Again, see the answer of J. Murray. In many instances, however, one gets away with talking about functions instead of equivalence classes. $\endgroup$ Commented Mar 5, 2023 at 19:29
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The correct answer to your question has already been posted, but I wanted to supplement that by noting a particular misconception:

If we want to apply this idea to the problem of a quantum particle in a box (infinite square well), then the wavefunctions representing all possible states of the system are both square-integrable and vanish at endpoints.

The Hilbert space for a particle in a box of length $a$ is $L^2\big([0,a]\big)$. There is no requirement that the wavefunctions vanish at the endpoints, and such a requirement in fact cannot even be imposed in the first place for two reasons.

  1. Despite the typical description, $L^2\big([0,a]\big)$ is not the space of square-integrable functions from $[0,a]\rightarrow \mathbb C$. Rather, it is the space of equivalence classes of such functions under the equivalence relation $\psi\sim \phi \iff \psi = \phi$ almost everywhere. Two functions which differ by their values at a single point represent the same element of $L^2\big([0,a]\big)$, and as a result we cannot impose any constraints which require wavefunctions to take a specific value at individual points, unless we also impose at least the requirement of continuity (or something stronger, like differentiability).

  2. Part of the definition of a Hilbert space is that it be topologically complete. This means that every Cauchy sequence $\{\psi_n\}$ must converge to a limit which is contained in the Hilbert space. However, notice that $$\psi_n(x) = \left[\frac{2x(a-x)}{a}\right]^{1/n}$$ is a Cauchy sequence, and that it converges (in the topology defined by the norm) to $\psi(x) = 1$: enter image description here

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