Notes on Lewin Ch1: Introduction to Quantum Mechanics

Important notation

• $⟨𝑓\text{,}𝑔⟩≔\int \overline{𝑓}𝑔$, so it is antilinear in the first index and linear in the second.
• $⟨𝐴⟩≔⟨𝜓\text{,}𝐴𝜓⟩$.

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

The reader was asked to verify Ehrenfest’s relations, which states that $\frac{d}{d𝑡}⟨𝑥⟩=⟨𝑝⟩$ and $\frac{d}{d𝑡}⟨𝑝⟩=-⟨{𝑉}^{\prime }⟩$. This comes out of the commutator relation $[𝐻,\text{x}]=𝑖𝑃$ and $[𝐻,𝑃]=-𝑖{𝑉}^{\prime }$. For the first, we have that $[𝑉,\text{x}]=0$, 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.

  $$[𝑋,𝑃]=\text{i}\hslash .$$

  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 $tr(𝐴𝐵)=tr(𝐵𝐴)$.

• For the quantum hydrogen atom, the Hamiltonian is $𝐻=-\mathrm{\Delta }-\frac{1}{|𝑥|}$ and the corresponding energy is

  $$\mathrm{ℰ︀}(𝜓)={\int }_{{ℝ}^3}{|\nabla 𝜓|}^{2}d𝑥-{\int }_{{ℝ}^3}\frac{{|𝜓|}^2}{|𝑥|}d𝑥$$

• Proposition 1.1 is that $\mathrm{ℰ︀}(𝜓)\ge 𝐶$ for $𝜓\in {𝐻}^1$ with $\Vert 𝜓{\Vert }_{{𝐿}^2}=1$, and some $𝐶<0$. The proof: Sobolev embedding in dimension 3 is ${𝐻}^1({ℝ}^3)\subset {𝐿}^6({ℝ}^3)$, 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 $\frac{1}{|}𝑥|$ to bound the near part by ${𝐶}_{𝑟}\Vert 𝜓{\Vert }_{{𝐿}^6}^2$ and the far part by $\frac{1}{𝑟}\Vert 𝜓{\Vert }_{{𝐿}^2}^2$. Choosing $𝑟$ 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 ${𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$ 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, ${𝜓}_{𝑛}\stackrel{{𝐻}^1}{\rightharpoonup }𝜓$ implies by Rellich-Kondrachov that ${𝜓}_{𝑛}\stackrel{{𝐿}^2(𝐾)}{\to }𝜓$, and then interpolating we get strong convergence of ${𝜓}_{𝑛}$ in ${𝐿}^{𝑟}(𝐾)$ for $𝑟<2{𝑝}^{\prime }$, i.e. strong convergence of $|{𝜓}_{𝑛}{|}^2$ in ${𝐿}^{𝑟}(𝐾)$ for $𝑟<2{𝑝}^{\prime }$. Then, since $|{𝜓}_{𝑛}{|}^2$ is bounded in ${𝐿}^{{𝑝}^{\prime }}$, we get weak convergence along a subsequence of $|{𝜓}_{𝑛}{|}^2$ in ${𝐿}^{{𝑝}^{\prime }}$ (testing against $𝑓\in {𝐶}_{𝑐}^{\infty }({ℝ}^{𝑑})$ 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 $𝑉$ into ${𝑉}_{+}+{𝑉}_{-}$ 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 $𝐽$ is wlsc if ${𝑓}_{𝑛}\rightharpoonup 𝑓$ implies $𝐽(𝑓)\le {lim\hspace{0.17em}inf}_{𝑛\to \infty }𝐽({𝑓}_{𝑛})$.

Exercises

1. 1. $𝑋+𝑌$ is Banach: this is actually routine so I will not write the details.
   2. Show the dual of $𝑋+𝑌$ is ${𝑋}^{\prime }\cap {𝑌}^{\prime }$ argument is standard.

   3. Question is to show that for $𝑓\in {𝐿}^{𝑝}+{𝐿}^{\infty }$, TFAE:

      1. $𝑓\in {𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$
      2. $|\{|𝑓|>𝜂\}|<\infty$ for all $𝜂>0$.
      3. ${lim}_{𝑅\to \infty }{\Vert {\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}𝑓\Vert }_{{𝐿}^{𝑝}+{𝐿}^{\infty }}=0.$

      Proof of $\text{(i)}⟹\text{(ii)}$: By definition we can find a decomposition $𝑓={𝑓}_{𝑝}+{𝑓}_{𝜂}$ with ${\Vert {𝑓}_{𝜂}\Vert }_{{𝐿}^{\infty }}<\frac{𝜂}{2}$. Then $𝜇(|𝑓|>𝜂)=𝜇(|{𝑓}_{𝑝}|>𝜂)\le \frac{{\Vert {𝑓}_{𝑝}\Vert }_{{𝐿}^{𝑝}}^{𝑝}}{{𝜂}^{𝑝}}$ just by Markov’s inequality, so it is finite.

      Proof of $\text{(ii)}⟹\text{(iii)}$: Let $𝑓={𝑓}_{𝑝}+{𝑓}_{\infty }$ be a decomposition of $𝑓$ with ${𝑓}_{𝑝}\in {𝐿}^{𝑝}$ and ${𝑓}_{\infty }\in {𝐿}^{\infty }$. Fix $𝜀>0$. We need to show that for $𝑅$ large enough, ${\Vert {\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}𝑓\Vert }_{{𝐿}^{𝑝}+{𝐿}^{\infty }}<𝜀$. We can split ${\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}𝑓=\left({\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}{𝑓}_{𝑝}+{\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}{𝑓}_{\infty }{\mathrm{𝟙}}_{|{𝑓}_{\infty }|>\frac{𝜀}{2}}\right)+{\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}{𝑓}_{\infty }{\mathrm{𝟙}}_{|{𝑓}_{\infty }|\le \frac{𝜀}{2}}$, and the first term is in ${𝐿}^{𝑝}$ and the second term is in ${𝐿}^{\infty }$ with norm at most $\frac{𝜀}{2}$. Since ${𝑓}_{𝑝}\in {𝐿}^{𝑝}$, we can choose $𝑅$ large enough so that ${\Vert {\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}{𝑓}_{𝑝}\Vert }_{{𝐿}^{𝑝}}<\frac{𝜀}{2}$, and then we are done.

      Proof of $\text{(iii)}⟹\text{(i)}$: Let $𝜀>0$. We need to show that we can write $𝑓={𝑓}_{𝑝}+{𝑓}_{\infty }$ with ${𝑓}_{𝑝}\in {𝐿}^{𝑝}$ and ${\Vert {𝑓}_{\infty }\Vert }_{{𝐿}^{\infty }}<𝜀$. By assumption we can find $𝑅$ large enough so that ${\Vert {\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}𝑓\Vert }_{{𝐿}^{𝑝}+{𝐿}^{\infty }}<𝜀$. So we can write ${\mathrm{𝟙}}_{{𝐵}_{𝑅}^{𝑐}}𝑓={𝑔}_{𝑝}+{𝑔}_{\infty }$ with ${𝑔}_{𝑝}\in {𝐿}^{𝑝}$ and ${\Vert {𝑔}_{\infty }\Vert }_{{𝐿}^{\infty }}<\frac{𝜀}{2}$. Then we can write $𝑓=({\mathrm{𝟙}}_{{𝐵}_{𝑅}}𝑓+{𝑔}_{𝑝})+{𝑔}_{\infty }$, which provides the required decomposition.

   4. Q: Let ${ℎ}_{𝑛},ℎ\in {𝐿}^{\infty },{ℎ}_{𝑛}\stackrel{{𝐿}^{\infty }}{\to }ℎ$ and suppose $𝜇(|{ℎ}_{𝑛}|>𝜂)<\infty$ for all $𝜂>0$. Show that $𝜇(|ℎ|>𝜂)<\infty$ for all $𝜂>0$. Proof: Let $𝜂>0$. By assumption there is an $𝑛$ such that ${\Vert {ℎ}_{𝑛}-ℎ\Vert }_{{𝐿}^{\infty }}<\frac{𝜂}{2}$. Then $𝜇(|ℎ|>𝜂)\le 𝜇\left(|{ℎ}_{𝑛}|>\frac{𝜂}{2}\right)$, which is finite by assumption.

   5. Show that ${𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$ is a closed subspace of ${𝐿}^{𝑝}+{𝐿}^{\infty }$. Proof: Let ${𝑓}_{𝑛}\in {𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$ be a sequence converging to $𝑓\in {𝐿}^{𝑝}+{𝐿}^{\infty }$. We need to show that $𝑓\in {𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$. We can decompose ${𝑓}_{𝑛}={𝑓}_{{𝑝}_{𝑛}}+{𝑓}_{{\infty }_{𝑛}}$ with ${𝑓}_{{𝑝}_{𝑛}}\in {𝐿}^{𝑝}$ and ${\Vert {𝑓}_{{\infty }_{𝑛}}\Vert }_{{𝐿}^{\infty }}<\frac{𝜀}{2}$. Since ${𝑓}_{𝑛}\to 𝑓$ in ${𝐿}^{𝑝}+{𝐿}^{\infty }$, we can find a decomposition $𝑓-{𝑓}_{𝑛}={𝑔}_{{𝑝}_{𝑛}}+{𝑔}_{{\infty }_{𝑛}}$ with ${𝑔}_{{𝑝}_{𝑛}}\in {𝐿}^{𝑝}$ and

      $${\Vert {𝑔}_{{\infty }_{𝑛}}\Vert }_{{𝐿}^{\infty }}<\underset{𝑓-{𝑓}_{𝑛}={𝑎}_{𝑝}+{𝑎}_{\infty }}{inf}{\Vert {𝑎}_{𝑝}\Vert }_{{𝐿}^{𝑝}}+{\Vert {𝑎}_{\infty }\Vert }_{{𝐿}^{\infty }}+\frac{𝜀}{2}<𝜀.$$

      Then we can write $𝑓=({𝑓}_{{𝑝}_{𝑛}}+{𝑔}_{{𝑝}_{𝑛}})+({𝑓}_{{\infty }_{𝑛}}+{𝑔}_{{\infty }_{𝑛}})$, which provides the required decomposition.

   6. Q: show that the closure of ${𝐶}_{𝑐}^{\infty }$ in ${𝐿}^{𝑝}+{𝐿}^{\infty }$ is ${𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$. Proof: by the previous question we know that $\overline{{𝐶}_{𝑐}^{\infty }}\subseteq {𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$. For the reverse inclusion, just find a negligible decomposition of $𝑓\in {𝐿}^{𝑝}+{𝐿}_{𝜀}^{\infty }$, drop the negligible part and approximate the ${𝐿}^{𝑝}$ part by a compactly supported function.

2. (Hardy’s inequality)

   1. Let $𝑑\ge 3$. Show that for all $𝜓\in {𝐻}^1({ℝ}^{𝑑})$, we have ${\int }_{{ℝ}^{𝑑}}\frac{{|𝜓|}^2}{{|𝑥|}^2}d𝑥\le \frac{4}{{(𝑑-2)}^2}{\int }_{{ℝ}^{𝑑}}{|\nabla 𝜓|}^{2}d𝑥$ by expanding the square ${\int }_{{ℝ}^{𝑑}}{\left|\nabla 𝜓+𝛼\frac{𝑥}{{|𝑥|}^2}𝜓\right|}^{2}d𝑥$ and optimizing over $𝛼$. Proof:

      $$0\le {\int }_{{ℝ}^{𝑑}}{\left|\nabla 𝜓+𝛼\frac{𝑥}{{|𝑥|}^2}𝜓\right|}^{2}d𝑥={\int }_{{ℝ}^{𝑑}}{|\nabla 𝜓|}^{2}d𝑥+{𝛼}^2{\int }_{{ℝ}^{𝑑}}\frac{{|𝜓|}^2}{{|𝑥|}^2}d𝑥+2𝛼{\int }_{{ℝ}^{𝑑}}\Re \left(\overline{𝜓}\frac{𝑥}{{|𝑥|}^2}\cdot \nabla 𝜓\right)d𝑥.$$

      Then we write $\nabla |𝜓{|}^2=\Re (2\overline{𝜓}\nabla 𝜓)$ and integrate by parts to see

      $${\int }_{{ℝ}^{𝑑}}\Re \left(\overline{𝜓}\frac{𝑥}{{|𝑥|}^2}\cdot \nabla 𝜓\right)d𝑥={\int }_{{ℝ}^{𝑑}}\frac{𝑥}{{|𝑥|}^2}\cdot \nabla |𝜓{|}^{2}d𝑥=-{\int }_{{ℝ}^{𝑑}}\nabla \cdot \left(\frac{𝑥}{{|𝑥|}^2}\right)|𝜓{|}^{2}d𝑥=-(𝑑-2){\int }_{{ℝ}^{𝑑}}\frac{{|𝜓|}^2}{{|𝑥|}^2}d𝑥.$$

      This implies

      $$(𝛼(𝑑-2)-{𝛼}^2){\int }_{{ℝ}^{𝑑}}\frac{{|𝜓|}^2}{{|𝑥|}^2}d𝑥\le {\int }_{{ℝ}^{𝑑}}{|\nabla 𝜓|}^{2}d𝑥.$$

      By completing the square we see that the quadratic is maximized with value $\frac{{(𝑑-2)}^2}{4}$, which gives the result.

   2. Q is to show above result extends to ${𝐻}^1$ functions - standard density argument.

   3. Q is to use $𝑢(𝑥)=𝑓(|𝑥|)$ with $𝑓(𝑟)={𝑟}^{-𝛼}{1}_{𝑟\le 1}+(2-𝑟)1_{1\le 𝑟\le 2}$, where $𝛼<\frac{𝑑-2}{2}$ to show optimality of the above constant. Proof: For radial functions, ${\int }_{{ℝ}^{𝑑}}𝑔(|𝑥|)d𝑥=|{𝑆}^{𝑑-1}|{\int }_0^{\infty }𝑔(𝑟){𝑟}^{𝑑-1}d𝑟$ Hence

      $${\int }_{{ℝ}^{𝑑}}\frac{{|𝑢(𝑥)|}^2}{{|𝑥|}^2}d𝑥=|{𝑆}^{𝑑-1}|{\int }_0^1{𝑟}^{𝑑-3-2𝛼}d𝑟+{\int }_1^2{(2-𝑟)}^2{𝑟}^{𝑑-3}d𝑟=|{𝑆}^{𝑑-1}|\frac{1}{𝑑-2-2𝛼}+{𝐴}_{𝑑},$$

      where ${𝐴}_{𝑑}≔{\int }_1^2{(2-𝑟)}^2{𝑟}^{𝑑-3}d𝑟<\infty$. Also, ${𝑓}^{\prime }(𝑟)=-𝛼{𝑟}^{-𝛼-1}$ on $(0,1)$ and ${𝑓}^{\prime }(𝑟)=-1$ on $(1,2)$, so ${\int }_{{ℝ}^{𝑑}}{|\nabla 𝑢(𝑥)|}^{2}d𝑥=|{𝑆}^{𝑑-1}|{𝛼}^2{\int }_0^1{𝑟}^{𝑑-3-2𝛼}d𝑟+{\int }_1^2{𝑟}^{𝑑-1}d𝑟=|{𝑆}^{𝑑-1}|\frac{{𝛼}^2}{𝑑-2-2𝛼}+{𝐵}_{𝑑}$,

      where ${𝐵}_{𝑑}≔{\int }_1^2{𝑟}^{𝑑-1}d𝑟<\infty$. Therefore

      $$\frac{{\int }_{{ℝ}^{𝑑}}\frac{{|𝑢(𝑥)|}^2}{{|𝑥|}^2}d𝑥}{{\int }_{{ℝ}^{𝑑}}{|\nabla 𝑢(𝑥)|}^{2}d𝑥}=\frac{\frac{1}{𝑑-2-2𝛼}+{𝐴}_{𝑑}}{\frac{{𝛼}^2}{𝑑-2-2𝛼}+{𝐵}_{𝑑}}$$

      .

      Letting $𝛼\to \frac{𝑑-2}{2}$,

      $$\frac{{\int }_{{ℝ}^{𝑑}}\frac{{|𝑢(𝑥)|}^2}{{|𝑥|}^2}d𝑥}{{\int }_{{ℝ}^{𝑑}}{|\nabla 𝑢(𝑥)|}^{2}d𝑥}\to \frac{1}{{\left(\frac{𝑑-2}{2}\right)}^2}={\left(\frac{2}{𝑑-2}\right)}^2$$

      , which shows optimality of the constant.

   4. Q is to show the stability of the hydrogen atom - i.e. that $\mathrm{ℰ︀}(𝜓)\ge -𝐶$ for some $𝐶>0$. Proof: By Hardy’s inequality we have

      $$\mathrm{ℰ︀}(𝜓)={\int }_{{ℝ}^3}{|\nabla 𝜓|}^{2}d𝑥-{\int }_{{ℝ}^3}\frac{{|𝜓|}^2}{|𝑥|}d𝑥\ge \frac{1}{4}{\int }_{{ℝ}^3}\frac{{|𝜓|}^2}{{|𝑥|}^2}d𝑥-{\int }_{{ℝ}^3}\frac{{|𝜓|}^2}{|𝑥|}d𝑥.$$

      Then we complete the square ${𝑡}^2-4𝑡={(𝑡-2)}^2-4$ to obtain the bound

      $$\mathrm{ℰ︀}(𝜓)\ge -1{\int }_{{ℝ}^3}{|𝜓|}^{2}d𝑥=-1.$$

      Curiously worse than the result from Sobolev inequality.

3. (Particle confined in ${ℝ}^{𝑑}$) Let $𝑉$ be a real-valued measurable function such that ${𝑉}_{-}=max(-𝑉,0)\in {𝐿}^{𝑝}({ℝ}^{𝑑},ℝ)+{𝐿}^{\infty }({ℝ}^{𝑑},ℝ)$, where $𝑝$ satisfies (1.37) and ${𝑉}_{+}=max(𝑉,0)\in {𝐿}_{\text{loc}}^1({ℝ}^{𝑑})$ with ${𝑉}_{+}(𝑥)\to +\infty$ when $|𝑥|\to +\infty$. We consider the energy

   $$\mathrm{ℰ︀}(𝜓)={\int }_{{ℝ}^{𝑑}}{|\nabla 𝜓(𝑥)|}^{2}d𝑥+{\int }_{{ℝ}^{𝑑}}𝑉(𝑥){|𝜓(𝑥)|}^{2}d𝑥$$

   and the space $\mathrm{𝒱︀}≔\left\{𝜓\in {𝐻}^1({ℝ}^{𝑑}):\sqrt{{𝑉}_{+}}𝜓\in {𝐿}^2({ℝ}^{𝑑})\right\}$ equipped with the norm

   $${\Vert 𝜓\Vert }_{\mathrm{𝒱︀}}^2={\Vert 𝜓\Vert }_{{𝐻}^1({ℝ}^{𝑑})}^2+{\int }_{{ℝ}^{𝑑}}{𝑉}_{+}(𝑥){|𝜓(𝑥)|}^{2}d𝑥.$$
   1. Show that $\mathrm{𝒱︀}$ is complete. Proof: Let ${𝜓}_{𝑛}$ be a Cauchy sequence in $\mathrm{𝒱︀}$. Then ${𝜓}_{𝑛}$ is a Cauchy sequence in ${𝐻}^1({ℝ}^{𝑑})$, so there is a $𝜓\in {𝐻}^1({ℝ}^{𝑑})$ such that ${𝜓}_{𝑛}\stackrel{{𝐻}^1}{\rightharpoonup }𝜓$. As ${𝜓}_{𝑛}$ is bounded in $\mathrm{𝒱︀}$, $\sqrt{{𝑉}_{+}}{𝜓}_{𝑛}$ is bounded in ${𝐿}^2({ℝ}^{𝑑})$, so by Fatou’s lemma, ${\int }_{{ℝ}^{𝑑}}{𝑉}_{+}|𝜓{|}^2\le {lim\hspace{0.17em}inf}_{𝑛}{\int }_{{ℝ}^{𝑑}}{𝑉}_{+}|{𝜓}_{𝑛}{|}^2\le 𝐶<\infty$. It follows that $𝜓\in \mathrm{𝒱︀}$. So $\mathrm{𝒱︀}$ is a closed subspace of ${𝐻}^1$, hence complete.
   2. Show that $\mathrm{ℰ︀}$ is well defined and continuous on $\mathrm{𝒱︀}$. Proof: well-definedness follows immediately from the previous part, with the bound $\mathrm{ℰ︀}(𝜓)\le {\Vert 𝜓\Vert }_{\mathrm{𝒱︀}}^2$ and $|\mathrm{ℰ︀}(𝜓)|\le 𝐶{\Vert 𝜓\Vert }_{\mathrm{𝒱︀}}^2$ (using Sobolev inequality and the definition of $𝑝$). For the continuity let ${𝜓}_{𝑛}\stackrel{\mathrm{𝒱︀}}{\to }𝜓$ and let $𝑀={sup}_{𝑛}{\Vert {𝜓}_{𝑛}\Vert }_{\mathrm{𝒱︀}}$. Then
   $$\begin{array}{cc}|\mathrm{ℰ︀}({𝜓}_{𝑛})-\mathrm{ℰ︀}(𝜓)|& \le {\int }_{{ℝ}^{𝑑}}|\nabla {𝜓}_{𝑛}+\nabla 𝜓||\nabla {𝜓}_{𝑛}-\nabla 𝜓|+{\int }_{{ℝ}^{𝑑}}𝑉|{𝜓}_{𝑛}+𝜓||{𝜓}_{𝑛}-𝜓|\\ & \le {\Vert {𝜓}_{𝑛}+𝜓\Vert }_{\mathrm{𝒱︀}}{\Vert {𝜓}_{𝑛}-𝜓\Vert }_{\mathrm{𝒱︀}}\le (𝐶+2𝑀){\Vert {𝜓}_{𝑛}-𝜓\Vert }_{\mathrm{𝒱︀}}\to 0.\end{array}$$
   3. Show that $𝐼=inf\left\{\mathrm{ℰ︀}(𝜓):𝜓\in \mathrm{𝒱︀},{\int }_{{ℝ}^{𝑑}}{|𝜓|}^2=1\right\}$ is finite. Proof: this is immediate from the case given in lectures by just dropping the positive part of $𝑉$.

   4. Show that $\mathrm{𝒱︀}$ is compactly embedded into ${𝐿}^2({ℝ}^{𝑑})$. Proof: As a subspace of ${𝐻}^1$, we have local compactness, but we have more due to ${𝑉}_{+}$. Let ${\Vert {𝑣}_{𝑛}\Vert }_{\mathrm{𝒱︀}}\le 𝐶$. For any $𝑅$, we have up to a subsequence that ${𝑣}_{𝑛}{\mathrm{𝟙}}_{{𝐵}_{𝑅}(0)}$ converges strongly in ${𝐿}^2$, which defines a limit $𝑣\in {𝐿}_{\text{loc}}^2$. Let ${𝐾}_{𝑅}={inf}_{{𝐵}_{𝑅(0)}^{𝑐}}𝑉\underset{𝑅\to \infty }{\to }\infty$. Then from ${(𝑎+𝑏)}^2\le 2{𝑎}^2+2{𝑏}^2$, we get the bound

      $$4𝐶>{\int }_{|𝑥|>𝑅}𝑉{|{𝑣}_{𝑛}-𝑣|}^2>{𝐾}_{𝑅}{\Vert {𝑣}_{𝑛}-𝑣\Vert }_{{𝐿}^2({𝐵}_{𝑅}{(0)}^{𝑐})}$$

      We can combine these as follows. Let $𝜀>0$. Choose $𝑅$ so that $\frac{𝐶}{{𝐾}_{𝑅}}<\frac{𝜀}{2}$. Then up to a subsequence, we get that for large enough $𝑛$, ${\Vert {𝑣}_{𝑛}-𝑣\Vert }_{{𝐿}^2({ℝ}^{𝑑})}\le \frac{𝜀}{2}+\frac{𝐶}{{𝐾}_{𝑅}}=𝜀$, whence the result.

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

      Proof: As in the case provided in the text, it is clear that $\mathrm{ℰ︀}$ is coercive (in the sense defined in the text, i.e. subsets of bounded $\mathrm{ℰ︀}$ are bounded in $\mathrm{𝒱︀}\subset {𝐻}^1$ and wlsc. So if we take a minimizing sequence in $\mathrm{𝒱︀}$ with unit ${𝐿}^2$ norm, we find (up to a subsequence) that it weakly converges to $𝜓\in \mathrm{𝒱︀}$, with ${\Vert 𝜓\Vert }_{{𝐿}^2}\le 1$. But as $\mathrm{𝒱︀}⋐{𝐿}^2$, we find in fact strong convergence in ${𝐿}^2$, so in fact ${\Vert 𝜓\Vert }_{{𝐿}^2}=1$ 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 ${𝑉}_{+}\in {𝐿}_{\text{loc}}^1({ℝ}^{𝑑})$. 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 $𝜓$, locally.

   7. We now consider the harmonic oscillator $𝑉(𝑥)={𝜔}^2{|𝑥|}^2$ which corresponds to attaching our quantum particle to a spring nailed at the origin, $𝜔$ being related to the stiffness of the spring.

      a. Show that

      $$\mathrm{ℰ︀}(𝜓)={\int }_{{ℝ}^{𝑑}}{|\nabla 𝜓(𝑥)+𝜔𝑥𝜓(𝑥)|}^{2}d𝑥+𝜔𝑑{\int }_{{ℝ}^{𝑑}}{|𝜓(𝑥)|}^{2}d𝑥$$

      for all $𝜓\in \mathrm{𝒱︀}$. 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 $𝐼=𝜔𝑑$ and that the minimizers are all of the form

      $$𝜓(𝑥)={\left(\frac{𝜔}{𝜋}\right)}^{\frac{𝑑}{4}}{𝑒}^{𝑖𝜃}{𝑒}^{-𝜔\frac{{|𝑥|}^2}{2}},𝜃\in ℝ.$$

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