Notes on Lewin Ch1: Introduction to Quantum Mechanics

Important notation

• , so it is antilinear in the first index and linear in the second.
• .

Also - as I am playing on my own I will try to use (note the upright x) to indicate the multiplication operator by .

The reader was asked to verify Ehrenfest’s relations, which states that and . This comes out of the commutator relation and . For the first, we have that , so

For the second,

• Quote: “Heisenberg [Hei25] started from the axiom that, in dimension .d=1, the position and momentum are described by two self-adjoint operators X and P satisfying the commutation relation.

  Such operators do not exist in finite dimension (to see this, take the trace) ”

  • minor elaboration: The trace of any commutator of matrices is zero as .

• For the quantum hydrogen atom, the Hamiltonian is and the corresponding energy is

• Proposition 1.1 is that for with , and some . The proof: Sobolev embedding in dimension 3 is , whic lower bounds the first term. Splitting the second term into near and far parts, we can use Holder’s inequality and the integrability of to bound the near part by and the far part by . Choosing appropriately gives the result.

  • curiously the next result is a proof that the infimum is attained without this stability result. Presumably this is because the author wanted to show the proof, not the result.
• The “negligible at infinity” spaces are defined, properties proven in exercises
• In Lemma 1.10 the strong convergence is not proven explicitly but it is simple (just Holder). For the weak convergence, implies by Rellich-Kondrachov that , and then interpolating we get strong convergence of in for , i.e. strong convergence of in for . Then, since is bounded in , we get weak convergence along a subsequence of in (testing against and the strong local convergence to identify the limit). Since this is true for arbitrary subsequences, we have weak convergence along the full sequence.
• In Corollary 1.11 there is no mentioned split of into but the remark sort of implies it, and anyway it is obvious from the proof.
• The (sequential) wlsc is not written out separately from the proof. it is the following - we say is wlsc if implies .

Exercises

1. 1. is Banach: this is actually routine so I will not write the details.
   2. Show the dual of is argument is standard.

   3. Question is to show that for , TFAE:

      2. for all .

      Proof of : By definition we can find a decomposition with . Then just by Markov’s inequality, so it is finite.

      Proof of : Let be a decomposition of with and . Fix . We need to show that for large enough, . We can split , and the first term is in and the second term is in with norm at most . Since , we can choose large enough so that , and then we are done.

      Proof of : Let . We need to show that we can write with and . By assumption we can find large enough so that . So we can write with and . Then we can write , which provides the required decomposition.

   4. Q: Let and suppose for all . Show that for all . Proof: Let . By assumption there is an such that . Then , which is finite by assumption.

   5. Show that is a closed subspace of . Proof: Let be a sequence converging to . We need to show that . We can decompose with and . Since in , we can find a decomposition with and

      Then we can write , which provides the required decomposition.

   6. Q: show that the closure of in is . Proof: by the previous question we know that . For the reverse inclusion, just find a negligible decomposition of , drop the negligible part and approximate the part by a compactly supported function.

2. (Hardy’s inequality)

   1. Let . Show that for all , we have by expanding the square and optimizing over . Proof:

      Then we write and integrate by parts to see

      This implies

      By completing the square we see that the quadratic is maximized with value , which gives the result.

   2. Q is to show above result extends to functions - standard density argument.

   3. Q is to use with , where to show optimality of the above constant. Proof: For radial functions, Hence

      where . Also, on and on , so ,

      where . Therefore

      .

      Letting ,

      , which shows optimality of the constant.

   4. Q is to show the stability of the hydrogen atom - i.e. that for some . Proof: By Hardy’s inequality we have

      Then we complete the square to obtain the bound

      Curiously worse than the result from Sobolev inequality.

3. (Particle confined in ) Let be a real-valued measurable function such that , where satisfies (1.37) and with when . We consider the energy

   and the space equipped with the norm

   1. Show that is complete. Proof: Let be a Cauchy sequence in . Then is a Cauchy sequence in , so there is a such that . As is bounded in , is bounded in , so by Fatou’s lemma, . It follows that . So is a closed subspace of , hence complete.
   2. Show that is well defined and continuous on . Proof: well-definedness follows immediately from the previous part, with the bound and (using Sobolev inequality and the definition of ). For the continuity let and let . Then
   3. Show that is finite. Proof: this is immediate from the case given in lectures by just dropping the positive part of .

   4. Show that is compactly embedded into . Proof: As a subspace of , we have local compactness, but we have more due to . Let . For any , we have up to a subsequence that converges strongly in , which defines a limit . Let . Then from , we get the bound

      We can combine these as follows. Let . Choose so that . Then up to a subsequence, we get that for large enough , , whence the result.

   5. Deduce that is attained and write the equation satisfied by any minimizer.

      Proof: As in the case provided in the text, it is clear that is coercive (in the sense defined in the text, i.e. subsets of bounded are bounded in and wlsc. So if we take a minimizing sequence in with unit norm, we find (up to a subsequence) that it weakly converges to , with . But as , we find in fact strong convergence in , so in fact as needed.

      As for the equation, it comes from showing that the energy has zero derivative at the minimizer along a path of admissible functions as in the text. One gets the Schrödinger equation.

   Haven’t solved the below yet…

   6. Show the uniqueness of the minimizer up to a phase when . You can follow the arguments of Sect. 1.6 and use [LL01, Thm. 9.10] instead of (1.63) to show the strict positivity of , locally.

   7. We now consider the harmonic oscillator which corresponds to attaching our quantum particle to a spring nailed at the origin, being related to the stiffness of the spring.

      a. Show that

      for all . This formula is the equivalent, for the harmonic oscillator, of the relation (1.28) for the hydrogen atom and of the ground state resolution (1.64).

      b. Deduce that and that the minimizers are all of the form

      c. Show Heisenberg’s inequality (1.21) by optimizing over .