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− | ==Part a)==
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− | Let \( \Phi_0(x) = e^{-\frac{1}{2}x^2} \) and define \( \Phi_n(x) = (-1)^n e^{\frac{1}{2}x^2}\left(\left(\frac{d}{dx}\right)^n e^{-x^2}\right) \). Show that \( \hat \Phi_n(k) = C\Phi_n(k)\), with \( C = C(n) \in \mathbb{C} \).
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− | ===Solution=== | + | == Problem 12 == |
| + | (by Madiso) |
| | | |
− | We show this by proving that \( \Phi_n(x) \) is an eigenfunction of the Fourier-transform with eigenvalue \((-i)^n \), i.e. \( \hat \Phi_n(k) = (-i)^n\Phi_n(k).\) | + | We consider the Hamiltonian of a 1D fermionic oscillator |
| + | $$ |
| + | H_F = -i\omega\psi_1\psi_2 \quad (1), |
| + | $$ |
| + | where the anti-commutator of the fermionic wave functions is given by |
| + | $$ |
| + | \{\psi_i,\psi_j\} = \hbar\delta_{ij} \quad (2) |
| + | $$ |
| + | We introduce the lowering and rising operators |
| + | $$ |
| + | \alpha = \frac{1}{\sqrt{2\hbar}} \left( \psi_1 - i\psi_2 \right), \quad \alpha^{\dagger} = \frac{1}{\sqrt{2\hbar}} \left( \psi_1 + i\psi_2 \right) \quad (3) |
| + | $$ |
| | | |
− | ''Proof.''
| + | === Part a) === |
| + | Show that \( \{\alpha,\alpha^\dagger\} = 1 \), \( \{\alpha,\alpha\} = \{\alpha^\dagger,\alpha^\dagger\} = 0 \) and \( \alpha^2 = \left( \alpha^\dagger \right)^2 = 0 \). |
| | | |
− | We first check the case \( \hat \Phi_0(k) \) (you don't have to know this calculation, we did that in Series 8, ex. 3 \( \color{red}{see \: discussion} \)) where we complete the square in the exponent by using the substitution \( \eta \equiv \frac{x + ik}{\sqrt{2}} \):
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− | $$ \hat \Phi_0(k) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R} } e^{-\frac{1}{2}x^2} e^{-ikx} dx = \frac{\sqrt{2}}{\sqrt{2\pi}} \int_{\mathbb{R} } e^{-\eta^2 - \frac{k^2}{2}} d\eta = e^{-\frac{1}{2} k^2} = \Phi_0(k) \tag{3.14} $$
| + | === Part b) === |
| + | Show that \( H_F = \hbar\omega\left(\alpha^\dagger\alpha - \frac{1}{2} \right) \). |
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− | For the general case:
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| | | |
− | $$ \sqrt{2\pi} \hat \Phi_n(k) = \int_{\mathbb{R} } (-1)^n e^{\frac{1}{2}x^2} \Big( (\frac{d}{dx})^n e^{-x^2} \Big) e^{-ikx} dx = \int_{\mathbb{R} } e^{-x^2} \Big( \frac{d}{dx} \Big)^n e^{\frac{1}{2}x^2 - ikx} dx $$
| + | ==Option One== |
| | | |
− | where we partially integrated. The first term of each partial integration is zero since:
| + | ==== Solution Part a) Writing it out==== |
| | | |
− | $$ \left| \left( \frac{d}{dx} \right)^m e^{\frac{1}{2}x^2 - ikx} \left( \left(\frac{d}{dx}\right)^l e^{-x^2} \right) \right| = \left| p(x) e^{-\frac{1}{2}x^2-ikx}\right| = \left| p(x)\right| e^{-\frac{1}{2}x^2} $$ | + | We start by using (3) |
| + | $$ |
| + | \begin{align} |
| + | \{\alpha,\alpha^\dagger\} &= \frac{1}{2\hbar} \{\left( \psi_1 - i\psi_2 \right), \left( \psi_1 + i\psi_2 \right) \} \\ |
| + | &= \frac{1}{2\hbar} \left[ (\psi_1\psi_1 + \psi_2\psi_2 + i\psi_1\psi_2 - i\psi_2\psi_1) + (\psi_1\psi_1 + \psi_2\psi_2 - i\psi_1\psi_2 + i\psi_2\psi_1) \right] \\ |
| + | &= \frac{1}{2\hbar} (2\psi_1\psi_1 + 2\psi_2\psi_2) = \frac{1}{2\hbar} \left( \{\psi_1, \psi_1\} + \{\psi_2, \psi_2\} \right) \\ |
| + | \end{align} |
| + | $$ |
| | | |
− | where \( p(x) = \mathcal{O}(x^j) \) (using the Landau notation) is a (complex) polynomial in \( x \) and \(l, m, j \in \mathbb{N}, \) leading to
| + | Using (2) we get |
| | | |
− | $$ \left( \frac{d}{dx} \right)^m e^{\frac{1}{2}x^2 - ikx} \left( \left(\frac{d}{dx}\right)^l e^{-x^2} \right)\bigg \vert_{-\infty}^{\infty} = 0 $$ | + | $$ |
| + | \{\alpha,\alpha^\dagger\} = \frac{2\hbar}{2\hbar} = 1 \quad (4) |
| + | $$ |
| | | |
− | since the exponential goes to \( 0 \) faster than any polynomial goes to \( \infty \) for \( x \rightarrow \pm \infty: \) $$ \left| p(x) \right| e^{-\frac{1}{2}x^2} \bigg \vert_{-\infty} = \left| p(x) \right| e^{-\frac{1}{2}x^2} \bigg \vert_{\infty} = 0. $$
| + | Next we see, that |
| | | |
− | We then go on like this:
| + | $$ |
| + | \begin{align} |
| + | \{\alpha,\alpha\} &= \frac{1}{2\hbar} \{\left( \psi_1 - i\psi_2 \right), \left( \psi_1 - i\psi_2 \right) \} \\ |
| + | &= \frac{1}{2\hbar} \left[ \left( \psi_1\psi_1 - \psi_2\psi_2 - i\psi_1\psi_2 - i\psi_2\psi_1 \right) + \left( \psi_1\psi_1 - \psi_2\psi_2 - i\psi_1\psi_2 - i\psi_2\psi_1 \right) \right] \\ |
| + | &= \frac{1}{2\hbar} \left( \{\psi_1,\psi_1\} - \{\psi_2,\psi_2\} -2i\{\psi_1,\psi_2\} \right) = 0. \\ |
| + | \end{align} |
| + | $$ |
| + | Similarly we get |
| + | $$ |
| + | \begin{align} |
| + | \{\alpha^\dagger,\alpha^\dagger\} &= \frac{1}{2\hbar} \{\left( \psi_1 + i\psi_2 \right), \left( \psi_1 + i\psi_2 \right) \} \\ |
| + | &= \frac{1}{2\hbar} \left[ \left( \psi_1\psi_1 - \psi_2\psi_2 + i\psi_1\psi_2 + i\psi_2\psi_1 \right) + \left( \psi_1\psi_1 - \psi_2\psi_2 + i\psi_1\psi_2 + i\psi_2\psi_1 \right) \right] \\ |
| + | &= \frac{1}{2\hbar} \left( \{\psi_1,\psi_1\} - \{\psi_2,\psi_2\} + 2i\{\psi_1,\psi_2\} \right) = 0. \\ |
| + | \end{align} |
| + | $$ |
| | | |
− | $$ \sqrt{2\pi} \hat \Phi_n(k) = e^{\frac{1}{2}k^2} \int_{\mathbb{R} } e^{-x^2} \Big( \frac{d}{dx} \Big)^n e^{\frac{1}{2}(x - ik)^2} dx = i^n e^{\frac{1}{2}k^2} \int_{\mathbb{R} } e^{-x^2} \Big( \frac{d}{dk} \Big)^n e^{\frac{1}{2}(x - ik)^2} dx $$ | + | Thus it is clear that |
| + | $$ |
| + | \{\alpha,\alpha\} = 2\alpha^2 = 0 \Rightarrow \alpha^2 = 0 \quad (5) |
| + | $$ |
| + | and |
| + | $$ |
| + | \{\alpha^\dagger,\alpha^\dagger\} = 2\left(\alpha^\dagger\right)^2 = 0 \Rightarrow \left(\alpha^\dagger\right)^2 = 0 \quad (6) |
| + | $$ |
| | | |
− | where we completed the square in the exponent and then used the following lemma:
| + | ==== Solution Part b)==== |
| | | |
− | '''Lemma 1:''' \( \forall n \in \mathbb{Z}_{\geqslant 0}: i^n \Big( \frac{d}{dk} \Big)^n e^{\frac{1}{2}(x - ik)^2} = \Big( \frac{d}{dx} \Big)^n e^{\frac{1}{2}(x - ik)^2} \)
| + | Using (3) we can express \(\psi_1\) and \(\psi_2\) from \(\alpha\) and \(\alpha^\dagger\) |
| + | $$ |
| + | \alpha + \alpha^\dagger = \frac{2}{\sqrt{2\hbar}}\psi_1 \Rightarrow \psi_1 = \sqrt{\frac{\hbar}{2}}\left( \alpha + \alpha^\dagger \right) |
| + | $$ |
| | | |
− | ''Proof of Lemma 1:''
| + | $$ |
| + | \alpha - \alpha^\dagger = \frac{-i2}{\sqrt{2\hbar}}\psi_2 \Rightarrow \psi_2 = i\sqrt{\frac{\hbar}{2}}\left( \alpha - \alpha^\dagger \right) |
| + | $$ |
| + | Now we can put these into (1) and express \(H_F\) from \(\alpha\) and \(\alpha^\dagger\) |
| + | $$ |
| + | H_F = -i\omega\psi_1\psi_2 = (-i\omega)i\frac{\hbar}{2} \left( \alpha + \alpha^\dagger \right)\left( \alpha - \alpha^\dagger \right) |
| + | $$ |
| | | |
− | By induction.
| + | $$ |
| + | = \frac{\hbar\omega}{2} \left( \alpha^2 - \alpha\alpha^\dagger + \alpha^\dagger\alpha - \left(\alpha^\dagger\right)^2 \right) |
| + | $$ |
| + | Using results (4),(5) and (6) from section a), we get |
| + | $$ |
| + | H_F = \frac{\hbar\omega}{2} \left( -\left( 1 - \alpha^\dagger\alpha \right) + \alpha^\dagger\alpha \right) = \hbar\omega \left( \alpha^\dagger\alpha - \frac{1}{2} \right). |
| + | $$ |
| | | |
− | \( n = 0 \) : \( i^0 e^{\frac{1}{2}(x - ik)^2} = e^{\frac{1}{2}(x - ik)^2} \) is the trivial case.
| |
| | | |
− | \( n-1 \rightarrow n \) :
| + | ==Option Two== |
| | | |
− | $$ i^n \Big( \frac{d}{dk} \Big)^n e^{\frac{1}{2}(x - ik)^2} = i^{n-1} \Big( \frac{d}{dk} \Big)^{n-1} i (-i) (x-ik) e^{\frac{1}{2}(x - ik)^2} = i^{n-1} \Big( \frac{d}{dk} \Big)^{n-1} (x-ik) e^{\frac{1}{2}(x - ik)^2} = \\ = i^{n-1} \Big( \frac{d}{dk} \Big)^{n - 1} \frac{d}{dx} e^{\frac{1}{2}(x - ik)^2} = i^{n-1} \frac{d}{dx} \Big( \frac{d}{dk} \Big)^{n - 1} e^{\frac{1}{2}(x - ik)^2} $$
| + | First of all we show that the anticommutator is bilinear and symmetrical: |
| | | |
− | where for the last equality we used Schwarz's theorem. With the induction assumption everything follows. \( \square \)
| + | $$ |
| + | \{\mu (A+B),\nu (C+D)\} = \mu (A+B) \nu (C+D) + \nu (C+D) \mu (A+B) = \mu \nu ((AC+AD+BC+BD) + (CA+CB+DA+DB)) = \mu \nu (\{A,C\} + \{A,D\} + \{B,C\} + \{B,D\}) |
| + | $$ |
| + | $$ |
| + | \{A,B\} = AB + BA = BA + AB = \{B,A\} |
| + | $$ |
| | | |
− | ''Okay, ladies and gentlemen, listen carefully. Lemma 2 is incredibly boring to prove. We've found a shorter and more general way to do this. I've marked the subparts that divide in two versions separately.''
| + | ==== Solution Part a) Use bilinearity of the anticommutator==== |
| | | |
− | '''Lemma 2:''' Let \( f \in \mathcal{S}(\mathbb{R}) \) and \( \lambda (k) \in C^{\infty} \) be a function only depending on k. Then: \( \left( \frac{d}{dk} \right)^{n} \lambda (k) \hat f(k) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \left( \frac{d}{dk} \right)^n \lambda (k) f(x) e^{-ikx} dx \).
| + | $$ |
| + | \begin{align} \\ |
| + | \{\alpha,\alpha^\dagger\} &= \frac{1}{2\hbar} \{( \psi_1 - i\psi_2 ), ( \psi_1 + i\psi_2 ) \} \\ |
| + | &= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} + i \{ \psi_1, \psi_2 \} - i \{ \psi_2, \psi_1 \} + \{ \psi_2, \psi_2 \}) \\ |
| + | &= \frac{1}{2\hbar} (\hbar + \hbar) \\ |
| + | &= 1 \\ |
| + | \end{align} |
| + | $$ |
| | | |
− | ''Proof.''
| + | $$ |
| + | \begin{align} \\ |
| + | \{\alpha,\alpha\} &= \frac{1}{2\hbar} \{( \psi_1 - i\psi_2 ), ( \psi_1 - i\psi_2 \} \\ |
| + | &= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} - i \{ \psi_1, \psi_2 \} - i \{ \psi_2, \psi_1 \} + \{ -i \psi_2, -i \psi_2 \}) \\ |
| + | &= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} -0 -0 - \{ \psi_2, \psi_2 \}) \\ |
| + | &= \frac{1}{2\hbar} (\hbar - \hbar) \\ |
| + | &= 0 \\ |
| + | \end{align} |
| + | $$ |
| | | |
− | \( \vdash \) : \( \mathcal{S}(\mathbb{R}) \subset L^1( \mathbb{R} ) \) | + | $$ |
| + | \begin{align} \\ |
| + | \{\alpha^\dagger\,\alpha^\dagger\} &= \frac{1}{2\hbar} \{( \psi_1 + i\psi_2 ), ( \psi_1 + i\psi_2 ) \} \\ |
| + | &= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} + i \{ \psi_1, \psi_2 \} + i \{ \psi_2, \psi_1 \} + \{ +i \psi_2, +i \psi_2 \}) \\ |
| + | &= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} + 0 + 0 - \{ \psi_2, \psi_2 \}) \\ |
| + | &= \frac{1}{2\hbar} (\hbar - \hbar) \\ |
| + | &= 0 \\ |
| + | \end{align} |
| + | $$ |
| | | |
− | Indeed, from Series 11, Ex. 1 we know:
| + | And thus as above: |
| + | $$ |
| + | \{\alpha,\alpha\} = 2\alpha^2 = 0 \Rightarrow \alpha^2 = 0 |
| + | $$ |
| + | and |
| + | $$ |
| + | \{\alpha^\dagger,\alpha^\dagger\} = 2\left(\alpha^\dagger\right)^2 = 0 \Rightarrow \left(\alpha^\dagger\right)^2 = 0 |
| + | $$ |
| | | |
− | $$ \left| f(x) \right| \leq \frac{C}{(1 + \vert x \vert^2)} $$
| |
| | | |
− | for some \( C \in \mathbb{R} \) and thus:
| + | ===Solution Part b)=== |
| | | |
− | $$ \int_{\mathbb{R}} \left| f(x) \right| dx \leq \int_{\mathbb{R}} \left| \frac{C}{(1 + \vert x \vert^2)} \right| dx < \infty $$ | + | $$ |
| + | \begin{align} |
| + | \alpha^\dagger \alpha\ &= \frac{1}{2\hbar} (\psi_1 + i \psi_2)(\psi_1 - i\psi_2) \\ |
| + | &= \frac{1}{2\hbar} (\psi_1^2 + \psi_2^2 - i \psi_1 \psi_2 + i \psi_2 \psi_1) \\ |
| + | &= \frac{1}{2\hbar} (\frac{1}{2} \{\psi_1, \psi_1 \} + \frac{1}{2} \{\psi_2, \psi_2 \} - i \psi_1 \psi_2 - i \psi_1 \psi_2 + i \psi_1 \psi_2 + i \psi_2 \psi_1) \\ |
| + | &= \frac{1}{2\hbar} (\hbar - 2i(\psi_1 \psi_2) + i\{ \psi_1, \psi_2 \} \\ |
| + | &= \frac{1}{2} - \frac{i}{\hbar}(\psi_1 \psi_2) + 0 \\ |
| + | \end{align} |
| + | $$ |
| + | $$ |
| + | \Rightarrow \psi_1 \psi_2 = -\frac{\hbar}{i}(\alpha^\dagger \alpha\ - \frac{1}{2} ) \quad (*) |
| + | $$ |
| | | |
− | <hr style="border:0;border-top:thin dashed #ccc;background:none;"/>
| + | Now, we insert \((*)\) into \( H_F \). |
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− | ''Version 1''
| + | $$ |
− | | + | |
− | Using the property
| + | |
− | | + | |
− | Now, we use an induction argument for \( \left( \frac{d}{dk} \right)^n \hat f(k) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \left( \frac{d}{dk} \right)^n f(x) e^{-ikx} dx \)
| + | |
− | | + | |
− | (well-defined since \( f \in \mathcal{S}(\mathbb{R}) \Rightarrow \hat f(k) \in \mathcal{S}(\mathbb{R}) \subset C^{\infty} \) from a lemma in the script):
| + | |
− | | + | |
− | \( n = 0 \) is the trivial case.
| + | |
− | | + | |
− | \( n - 1 \rightarrow n: \)
| + | |
− | | + | |
− | Let \( h_m \) be an arbitrary zero-sequence.
| + | |
− | | + | |
− | $$ \sqrt{2\pi} \left( \frac{d}{dk} \right)^{n} \hat f(k) = \sqrt{2\pi} \frac{d}{dk} \left( \frac{d}{dk} \right)^{n-1} \hat f(k) =^{\color{green}{*}} \lim_{m \rightarrow \infty} \int_{\mathbb{R}} \frac{1}{h_m} \left( \frac{d}{dk} \right)^{n-1} f(x) e^{ - ikx} \left( e^{-ixh_m} - 1 \right) dx $$
| + | |
− | | + | |
− | $$ \color{green}{*} \left( \left( \frac{d}{dk} \right)^{j} e^{ - ikx} \right) \bigg \vert_{k + h_m} = \left( p(x)e^{ - ikx} \right) \bigg \vert_{k + h_m} = p(x)e^{ - i(k+h_m)x} = \left( \frac{d}{dk} \right)^{j} e^{ - ikx} e^{ - ih_mx} $$
| + | |
− | | + | |
− | We then proceed:
| + | |
− |
| + | |
− | $$ \begin{align} \left| \frac{1}{h_m} \left( \frac{d}{dk} \right)^{n-1} f(x) e^{-ikx} \left( e^{-i h_m x} - 1 \right) \right| &= \left| \frac{2}{h_m} f(x) \sin \left( \frac{xh_m}{2} \right) \left( \frac{d}{dk} \right)^{n-1} e^{-ikx} \right| \leq \left| \frac{2}{h_m} f(x) \frac{xh_m}{2} \left( \frac{d}{dk} \right)^{n-1} e^{-ikx} \right| \\
| + | |
− | &= \left| x \cdot f(x) \left( \frac{d}{dk} \right)^{n-1} e^{-ikx} \right| = \left| p(x) \cdot f(x) \right| \in L^1
| + | |
− | \end{align}$$
| + | |
− | | + | |
− | where we used \( \vert \sin (x) \vert \leq \vert x \vert \) and \( p(x) \in \mathbb{C}[x] \). Using Lebesgue dominated convergence theorem, the induction proof is concluded.
| + | |
− | | + | |
− | <hr style="border:0;border-top:thin dashed #ccc;background:none;"/>
| + | |
− | | + | |
− | ''Version 2:''
| + | |
− | | + | |
− | Using the property from the script: \( \frac{d}{dk} \widehat{f} (k) = (-i) \cdot \widehat{\left( xf \right) }(k) \) we see easily that \( \left( \frac{d}{dk} \right)^n \widehat{f} (k) = \left( -i \right)^n \widehat{ \left( x^n \cdot f \right) } (k) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \left( \frac{d}{dk} \right)^n f(x) e^{-ikx} dx \) since \( \left( \frac{d}{dk} \right)^n e^{-ikx} = (-ix)^n \cdot e^{-ikx} \). ''If you like you may of course do an induction here but it seems clear enough to me.''
| + | |
− | | + | |
− | Note that \( x^{\alpha} \cdot f \in L^1 \) for any \( \alpha \in \{ 0, ..., n \} \) follows from \( f \in \mathcal{S}(\mathbb{R}) \).
| + | |
− | | + | |
− | <hr style="border:0;border-top:thin dashed #ccc;background:none;"/>
| + | |
− | | + | |
− | ''Here the fans of version 1 and version 2 have to work together again.''
| + | |
− | | + | |
− | We now use the generalized Leibniz-rule to show:
| + | |
− | | + | |
− | $$\begin{align}
| + | |
− | \left(\frac{d}{dk}\right)^{n} \lambda (k)\int_\mathbb{R} f(x) e^{-ikx}dx &= \sum_{j=0}^{n} \binom{n}{j}\left( \left( \frac{d}{dk} \right)^j \lambda (k) \right) \left( \frac{d}{dk} \right)^{n - j} \int_\mathbb{R} f(x) e^{-ikx} dx \\
| + | |
− | &= \sum_{j=0}^{n} \binom{n}{j} \left( \left( \frac{d}{dk} \right)^j \lambda (k) \right) \int_\mathbb{R} \left( \frac{d}{dk} \right)^{n - j} f(x) e^{-ikx} dx \\
| + | |
− | &= \sum_{j=0}^{n} \int_\mathbb{R} \binom{n}{j} \left( \left( \frac{d}{dk} \right)^j \lambda (k) \right) \left( \frac{d}{dk} \right)^{n - j} f(x) e^{-ikx} dx \\
| + | |
− | &= \int_\mathbb{R} \sum_{j=0}^{n} \binom{n}{j} \left( \left( \frac{d}{dk} \right)^j \lambda (k) \right) \left( \frac{d}{dk} \right)^{n - j} f(x) e^{-ikx} dx \\
| + | |
− | &= \int_\mathbb{R} \left( \frac{d}{dk} \right)^n \lambda (k) f(x) e^{-ikx} dx
| + | |
− | \end{align}$$
| + | |
− | | + | |
− | And the Lemma is proven. \( \square \)
| + | |
− | | + | |
− | From this Lemma follows directly for \( f(x) = e^{- \frac{x^2}{2} } \in \mathcal{S}(\mathbb{R}) \) and \( e^{-\frac{k^2}{2}} \in C^{\infty} \)
| + | |
− | | + | |
− | $$ \left( \frac{d}{dk} \right)^n e^{\frac{-k^2}{2}} \hat f(k) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \left( \frac{d}{dk} \right)^n e^{- \frac{k^2}{2} - x^2 + \frac{1}{2}x^2 - ikx} dx = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \left( \frac{d}{dk} \right)^n e^{- x^2 + \frac{1}{2} (x - ik)^2 } dx $$
| + | |
− | | + | |
− | We then finish the proof as follows:
| + | |
− | | + | |
− | $$ \sqrt{2\pi} \hat \Phi_n(k) = i^n e^{\frac{1}{2}k^2} \Big( \frac{d}{dk} \Big)^n \int_{\mathbb{R} } e^{-x^2 + \frac{1}{2}(x - ik)^2} dx = i^n \sqrt{2\pi} e^{\frac{1}{2}k^2} \Big( \frac{d}{dk} \Big)^n e^{-k^2} = (-i)^n \sqrt{2\pi} \Phi_n(k) $$
| + | |
− | | + | |
− | Where for the first equality we used Lemma 2 and for the second equation (3.14). \( \square \)
| + | |
− | | + | |
− | ===Aternative Solution (not proof-read yet) ===
| + | |
− | | + | |
− | claim: \( \hat \Phi_n(k) = (-i)^n\Phi_n(k)\)
| + | |
− | | + | |
− | proof by induction:
| + | |
− | | + | |
− | n = 0, using the Fundamental identity in the script (Fourier_SchwartzAdded 95):
| + | |
− | | + | |
− | $$ \hat \Phi_0(k) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R} } e^{-\frac{1}{2}x^2} e^{-ikx} dx = e^{-\frac{1}{2} k^2} = \Phi_0(k)$$
| + | |
− | | + | |
− | induction step: $$ \hat \Phi_{n+1}(k) =\frac{1}{\sqrt{2\pi}} \int_{\mathbb{R} } (-1)^{n+1} e^{\frac{1}{2}x^2} \Big( (\frac{d}{dx})^{n+1} e^{-x^2} \Big) e^{-ikx} dx $$
| + | |
− | | + | |
− | integrating by parts, where the additional term goes to zeros:
| + | |
− | | + | |
− | $$= - \frac{1}{\sqrt{2\pi}}(-1)^{n+1} \int_{\mathbb{R} } \Big((\frac{d}{dx})^{n+1}e^{-x^2}\Big) \frac{d}{dx}\Big(e^{\frac{1}{2}x^2} e^{-ikx}\Big) dx $$
| + | |
− | | + | |
− | $$= \frac{1}{\sqrt{2\pi}} (-1)^n \int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) x e^{-ikx} dx \ + \ \frac{1}{\sqrt{2\pi}} (-1)^{n+1}ik\int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) e^{-ikx} dx$$
| + | |
− | | + | |
− | $$\tag{$\ast$} = \frac{1}{\sqrt{2\pi}}(-1)^n \int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) \frac{1}{-i}\frac{d}{dk} e^{-ikx} dx \ - \ ik\hat\Phi_n(k)$$
| + | |
− | | + | |
− | taking the differatial outside, using again the mean value theorem and the dominated convergenz theorem. doing it the reverse way:
| + | |
− | | + | |
− | $$\frac{d}{dk} \int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) i e^{-ikx} dx = \lim\limits_{h \rightarrow 0} \int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) i \frac{e^{-ix(k+h)}-e^{-ixk}}{h} dx$$
| + | |
− | | + | |
− | (repressing the facotr \((-1)^n\))
| + | |
− | | + | |
− | $$= \lim\limits_{m \rightarrow \infty} \int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) i \frac{e^{-ix(k+h_m)}-e^{-ixk}}{h_m} dx = \lim\limits_{m \rightarrow \infty} \int_{\mathbb{R} } f_m$$
| + | |
− | | + | |
− | where \( \lim\limits_{m \rightarrow \infty} h_m = 0 \) an arbitary sequenze going to zero. from the mean value theorem we know that there is a \(\xi\) between 0 and \(h_m\), such that:
| + | |
− | | + | |
− | $$\frac{e^{-ix(k+h_m)}-e^{-ixk}}{h_m} = -ixe^{-ix(k+\xi)}$$
| + | |
− | | + | |
− | $$ \Rightarrow |f_m| \leq |x e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big)| = |p(x)e^{-\frac{1}{2}x^2}| \in L_1$$
| + | |
− | | + | |
− | where p(x) is a polynomial of order n+1. We also know that
| + | |
− | | + | |
− | $$\lim\limits_{m \rightarrow \infty} f_m= \lim\limits_{m \rightarrow \infty} e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) i\frac{e^{-ix(k+h_m)}-e^{-ixk}}{h_m} dx = e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) i(-ix)e^{-ixk}$$
| + | |
− | | + | |
− | Thus the dominated Convergenze Theorem applys and we can take the differantial inside.
| + | |
− | | + | |
− | $$\frac{d}{dk} \int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) i e^{-ikx} dx = \int_{\mathbb{R} } e^{\frac{1}{2}x^2} \Big((\frac{d}{dx})^{n}e^{-x^2}\Big) xe^{-ikx} dx$$
| + | |
− | | + | |
− | returning to the induction:
| + | |
− | | + | |
− | $$ (\ast) = i \frac{d}{dk}\hat\Phi_n(k) \ - \ ik\hat\Phi_n(k)$$
| + | |
− | | + | |
− | using the induction assumtion:
| + | |
− | | + | |
− | $$= i \frac{d}{dk}\Big( (-i)^n (-1)^n e^{\frac{1}{2}k^2} \Big( (\frac{d}{dk})^n
| + | |
− | e^{-k^2} \Big)\Big) \ - \ ik\hat\Phi_n(k)$$
| + | |
− | | + | |
− | $$= i (-i)^n (-1)^n \Big( k e^{\frac{1}{2}k^2} \Big( (\frac{d}{dk})^n
| + | |
− | e^{-k^2} \Big) + e^{\frac{1}{2}k^2} \Big( (\frac{d}{dk})^{n+1}
| + | |
− | e^{-k^2} \Big)\Big)\ - \ ik\hat\Phi_n(k)$$
| + | |
− | | + | |
− | $$= i (-i)^n (-1)^n e^{\frac{1}{2}k^2} \Big( (\frac{d}{dk})^{n+1}
| + | |
− | e^{-k^2} \Big) = (-i)^{n+1} \Phi_{n+1}(k) $$
| + | |
− | <p style="text-align:right;">\(\square\)</p>
| + | |
− | | + | |
− | | + | |
− | ==Part b) ==
| + | |
− | | + | |
− | Let \( \chi_{[a,b]} \) be the characteristic function, \( a, b \in \mathbb{R}, a < b \). Show explicitly, i.e. without using Plancherel, that
| + | |
− | | + | |
− | $$ \Vert \hat \chi_{[a,b]} \Vert_2^2 = b - a. $$
| + | |
− | | + | |
− | ''Hint:'' You may use without proof that
| + | |
− | | + | |
− | $$ \int_{0}^{\infty} \frac{\sin x}{x} dx = \frac{\pi}{2} $$
| + | |
− | | + | |
− | | + | |
− | ===Solution===
| + | |
− | | + | |
− | ''Nice to know:'' Plancherel's theorem states that for any function \( f \in L^2: \Vert \hat f \Vert_{2} = \Vert f \Vert_{2} \) where \( \Vert f \Vert_{2} = \left( \int_{\mathbb{R}} \vert f(x) \vert^2 dx \right)^{\frac{1}{2}} \)
| + | |
− | | + | |
− | But we show the identity like badasses by direct calculation.
| + | |
− | | + | |
− | We have: $$ \hat \chi_{[a,b]}(k) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \chi_{[a,b]}(x) e^{-ikx} dx = \frac{1}{\sqrt{2\pi}} \int_{a}^b e^{-ikx} dx$$
| + | |
− | | + | |
− | Now we use the substitution \( y \equiv \frac{2(x-a)}{b-a} -1 \) to scale the integral to the interval \( \left[ -1, 1 \right] \). Thus:
| + | |
− | | + | |
− | $$ | + | |
| \begin{align} | | \begin{align} |
− | \hat \chi_{[a,b]}(k) &= \frac{(b-a)}{2\sqrt{2\pi}} e^{-ik \left( \frac{a+b}{2} \right)}\int_{-1}^{1} e^{-ik \frac{b-a}{2} y} dy \\
| + | H_F &= -i \omega \psi_1 \psi_2 \\ |
− | &= \frac{(b-a)}{2\sqrt{2\pi}} e^{-ik \left( \frac{a+b}{2} \right)} \left( \frac{-2}{ik(b-a)} e^{-\frac{ik(b-a)}{2}y} \bigg \vert_{-1}^{1} \right) \\ | + | &\stackrel{(*)}{=} i \omega \frac{\hbar}{i}(\alpha^\dagger \alpha\ - \frac{1}{2} ) \\ |
− | &= \frac{1}{\sqrt{2\pi}} e^{-ik \left( \frac{a+b}{2} \right)} \frac{1}{ik} \left( - e^{-ik\frac{(b-a)}{2}} + e^{ik \frac{b-a}{2}} \right) \\
| + | &=\omega \hbar (\alpha^\dagger \alpha\ - \frac{1}{2} ) |
− | &= \frac{\sqrt{2}}{k\sqrt{\pi}} e^{-ik \left( \frac{a+b}{2} \right)} \sin \left( \frac{b-a}{2} k \right) | + | |
| \end{align} | | \end{align} |
| $$ | | $$ |
− |
| |
− |
| |
− |
| |
− | And so:
| |
− |
| |
− | $$ \Vert \hat \chi_{[a,b]}(k) \Vert_2^2 = \int_{\mathbb{R} } |\hat \chi_{[a,b]}(k)|^2 dk = \frac{2}{\pi} \int_{\mathbb{R} } \frac{\sin^2(\frac{b-a}{2}k)}{k^2} dk $$
| |
− |
| |
− | With the substitution \( \eta \equiv \frac{b-a}{2}k \) we get:
| |
− |
| |
− | $$ \Vert \hat \chi_{[a,b]}(k) \Vert_2^2 = \frac{(b-a)}{\pi} \int_{\mathbb{R} } \frac{\sin^2(\eta)}{\eta^2} d\eta $$
| |
− |
| |
− | It remains to evaluate the integral \( \int_{\mathbb{R} } \frac{\sin^2\left(\eta\right)}{\eta^2} d\eta \).
| |
− |
| |
− | By using the identity \( \sin^2 \left( x \right) = \frac{1}{2} \left( 1 - \cos \left( 2x \right) \right) \) and partial integration we have
| |
− |
| |
− | $$ \begin{align}
| |
− | \int_{\mathbb{R} } \frac{\sin^2\left(\eta\right)}{\eta^2} d\eta &= - \frac{1}{2\eta} \left( 1 - \cos \left( 2\eta \right) \right) \Bigg \vert_{- \infty}^{\infty} + \int_{\mathbb{R} } \frac{\sin \left(2\eta \right) }{\eta} d\eta \\
| |
− | &= 2 \int_{0}^{\infty} \frac{\sin \left(2\eta \right) }{\eta} d\eta \\
| |
− | &= 2 \int_{0}^{\infty} \frac{\sin\left( \xi \right)}{\xi} d\xi = \pi
| |
− | \end{align}$$
| |
− |
| |
− | where obviously \( \left| \frac{1}{2\eta} \left( 1 - \cos \left( 2\eta \right) \right) \right| \leq \left| \frac{1}{\eta} \right| \rightarrow 0 \) for \( \eta \rightarrow \pm \infty \).
| |
− |
| |
− | The result therefore is:
| |
− |
| |
− | $$ \Vert \hat \chi_{[a,b]}(k) \Vert_2^2 = b-a $$
| |
We consider the Hamiltonian of a 1D fermionic oscillator
$$
H_F = -i\omega\psi_1\psi_2 \quad (1),
$$
where the anti-commutator of the fermionic wave functions is given by
$$
\{\psi_i,\psi_j\} = \hbar\delta_{ij} \quad (2)
$$
We introduce the lowering and rising operators
$$
\alpha = \frac{1}{\sqrt{2\hbar}} \left( \psi_1 - i\psi_2 \right), \quad \alpha^{\dagger} = \frac{1}{\sqrt{2\hbar}} \left( \psi_1 + i\psi_2 \right) \quad (3)
$$
Show that \( \{\alpha,\alpha^\dagger\} = 1 \), \( \{\alpha,\alpha\} = \{\alpha^\dagger,\alpha^\dagger\} = 0 \) and \( \alpha^2 = \left( \alpha^\dagger \right)^2 = 0 \).
Show that \( H_F = \hbar\omega\left(\alpha^\dagger\alpha - \frac{1}{2} \right) \).
We start by using (3)
$$
\begin{align}
\{\alpha,\alpha^\dagger\} &= \frac{1}{2\hbar} \{\left( \psi_1 - i\psi_2 \right), \left( \psi_1 + i\psi_2 \right) \} \\
&= \frac{1}{2\hbar} \left[ (\psi_1\psi_1 + \psi_2\psi_2 + i\psi_1\psi_2 - i\psi_2\psi_1) + (\psi_1\psi_1 + \psi_2\psi_2 - i\psi_1\psi_2 + i\psi_2\psi_1) \right] \\
&= \frac{1}{2\hbar} (2\psi_1\psi_1 + 2\psi_2\psi_2) = \frac{1}{2\hbar} \left( \{\psi_1, \psi_1\} + \{\psi_2, \psi_2\} \right) \\
\end{align}
$$
$$
\begin{align}
\{\alpha,\alpha\} &= \frac{1}{2\hbar} \{\left( \psi_1 - i\psi_2 \right), \left( \psi_1 - i\psi_2 \right) \} \\
&= \frac{1}{2\hbar} \left[ \left( \psi_1\psi_1 - \psi_2\psi_2 - i\psi_1\psi_2 - i\psi_2\psi_1 \right) + \left( \psi_1\psi_1 - \psi_2\psi_2 - i\psi_1\psi_2 - i\psi_2\psi_1 \right) \right] \\
&= \frac{1}{2\hbar} \left( \{\psi_1,\psi_1\} - \{\psi_2,\psi_2\} -2i\{\psi_1,\psi_2\} \right) = 0. \\
\end{align}
$$
Similarly we get
$$
\begin{align}
\{\alpha^\dagger,\alpha^\dagger\} &= \frac{1}{2\hbar} \{\left( \psi_1 + i\psi_2 \right), \left( \psi_1 + i\psi_2 \right) \} \\
&= \frac{1}{2\hbar} \left[ \left( \psi_1\psi_1 - \psi_2\psi_2 + i\psi_1\psi_2 + i\psi_2\psi_1 \right) + \left( \psi_1\psi_1 - \psi_2\psi_2 + i\psi_1\psi_2 + i\psi_2\psi_1 \right) \right] \\
&= \frac{1}{2\hbar} \left( \{\psi_1,\psi_1\} - \{\psi_2,\psi_2\} + 2i\{\psi_1,\psi_2\} \right) = 0. \\
\end{align}
$$
Thus it is clear that
$$
\{\alpha,\alpha\} = 2\alpha^2 = 0 \Rightarrow \alpha^2 = 0 \quad (5)
$$
and
$$
\{\alpha^\dagger,\alpha^\dagger\} = 2\left(\alpha^\dagger\right)^2 = 0 \Rightarrow \left(\alpha^\dagger\right)^2 = 0 \quad (6)
$$
Using (3) we can express \(\psi_1\) and \(\psi_2\) from \(\alpha\) and \(\alpha^\dagger\)
$$
\alpha + \alpha^\dagger = \frac{2}{\sqrt{2\hbar}}\psi_1 \Rightarrow \psi_1 = \sqrt{\frac{\hbar}{2}}\left( \alpha + \alpha^\dagger \right)
$$
$$
\alpha - \alpha^\dagger = \frac{-i2}{\sqrt{2\hbar}}\psi_2 \Rightarrow \psi_2 = i\sqrt{\frac{\hbar}{2}}\left( \alpha - \alpha^\dagger \right)
$$
Now we can put these into (1) and express \(H_F\) from \(\alpha\) and \(\alpha^\dagger\)
$$
H_F = -i\omega\psi_1\psi_2 = (-i\omega)i\frac{\hbar}{2} \left( \alpha + \alpha^\dagger \right)\left( \alpha - \alpha^\dagger \right)
$$
$$
= \frac{\hbar\omega}{2} \left( \alpha^2 - \alpha\alpha^\dagger + \alpha^\dagger\alpha - \left(\alpha^\dagger\right)^2 \right)
$$
Using results (4),(5) and (6) from section a), we get
$$
H_F = \frac{\hbar\omega}{2} \left( -\left( 1 - \alpha^\dagger\alpha \right) + \alpha^\dagger\alpha \right) = \hbar\omega \left( \alpha^\dagger\alpha - \frac{1}{2} \right).
$$
$$
\{\mu (A+B),\nu (C+D)\} = \mu (A+B) \nu (C+D) + \nu (C+D) \mu (A+B) = \mu \nu ((AC+AD+BC+BD) + (CA+CB+DA+DB)) = \mu \nu (\{A,C\} + \{A,D\} + \{B,C\} + \{B,D\})
$$
$$
\{A,B\} = AB + BA = BA + AB = \{B,A\}
$$
$$
\begin{align} \\
\{\alpha,\alpha^\dagger\} &= \frac{1}{2\hbar} \{( \psi_1 - i\psi_2 ), ( \psi_1 + i\psi_2 ) \} \\
&= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} + i \{ \psi_1, \psi_2 \} - i \{ \psi_2, \psi_1 \} + \{ \psi_2, \psi_2 \}) \\
&= \frac{1}{2\hbar} (\hbar + \hbar) \\
&= 1 \\
\end{align}
$$
$$
\begin{align} \\
\{\alpha,\alpha\} &= \frac{1}{2\hbar} \{( \psi_1 - i\psi_2 ), ( \psi_1 - i\psi_2 \} \\
&= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} - i \{ \psi_1, \psi_2 \} - i \{ \psi_2, \psi_1 \} + \{ -i \psi_2, -i \psi_2 \}) \\
&= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} -0 -0 - \{ \psi_2, \psi_2 \}) \\
&= \frac{1}{2\hbar} (\hbar - \hbar) \\
&= 0 \\
\end{align}
$$
$$
\begin{align} \\
\{\alpha^\dagger\,\alpha^\dagger\} &= \frac{1}{2\hbar} \{( \psi_1 + i\psi_2 ), ( \psi_1 + i\psi_2 ) \} \\
&= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} + i \{ \psi_1, \psi_2 \} + i \{ \psi_2, \psi_1 \} + \{ +i \psi_2, +i \psi_2 \}) \\
&= \frac{1}{2\hbar} (\{ \psi_1, \psi_1 \} + 0 + 0 - \{ \psi_2, \psi_2 \}) \\
&= \frac{1}{2\hbar} (\hbar - \hbar) \\
&= 0 \\
\end{align}
$$
And thus as above:
$$
\{\alpha,\alpha\} = 2\alpha^2 = 0 \Rightarrow \alpha^2 = 0
$$
and
$$
\{\alpha^\dagger,\alpha^\dagger\} = 2\left(\alpha^\dagger\right)^2 = 0 \Rightarrow \left(\alpha^\dagger\right)^2 = 0
$$
$$
\begin{align}
\alpha^\dagger \alpha\ &= \frac{1}{2\hbar} (\psi_1 + i \psi_2)(\psi_1 - i\psi_2) \\
&= \frac{1}{2\hbar} (\psi_1^2 + \psi_2^2 - i \psi_1 \psi_2 + i \psi_2 \psi_1) \\
&= \frac{1}{2\hbar} (\frac{1}{2} \{\psi_1, \psi_1 \} + \frac{1}{2} \{\psi_2, \psi_2 \} - i \psi_1 \psi_2 - i \psi_1 \psi_2 + i \psi_1 \psi_2 + i \psi_2 \psi_1) \\
&= \frac{1}{2\hbar} (\hbar - 2i(\psi_1 \psi_2) + i\{ \psi_1, \psi_2 \} \\
&= \frac{1}{2} - \frac{i}{\hbar}(\psi_1 \psi_2) + 0 \\
\end{align}
$$
$$
\Rightarrow \psi_1 \psi_2 = -\frac{\hbar}{i}(\alpha^\dagger \alpha\ - \frac{1}{2} ) \quad (*)
$$
Now, we insert \((*)\) into \( H_F \).
$$
\begin{align}
H_F &= -i \omega \psi_1 \psi_2 \\
&\stackrel{(*)}{=} i \omega \frac{\hbar}{i}(\alpha^\dagger \alpha\ - \frac{1}{2} ) \\
&=\omega \hbar (\alpha^\dagger \alpha\ - \frac{1}{2} )
\end{align}
$$