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Inner Products of Energy Eigenstates for a 1-D Quantum Barrier

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Abstract

The features of the standard inner products between all the types of real and complex-energy solutions of the Schrödinger equation for 1-dimensional cut-off quantum potentials are worked out using a Gaussian regularization. A general Master Solution is introduced which describes any of the above solutions as particular cases. From it, a Master Inner Product is obtained which yields all the particular products. We show that the Outgoing and the Incoming Boundary Conditions fully determine the location of the momenta respectively in the lower and upper half complex plane even for purely imaginary momenta (anti-bound and bound solutions).

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References

  1. Gamow, G.: Zur Quantentheorie des AtomKernes. Z. Phys. 51, 204 (1928)

    Article  ADS  MATH  Google Scholar 

  2. Nicolaides, C.A.: Irreversibility in the framework of Hermitian and non-Hermitian treatments of resonance states. Lect. Notes Phys. 622, 357 (2003)

    Article  ADS  Google Scholar 

  3. Antoniou, I.E., Gadella, M.: Irreversibility, resonances and rigged Hilbert spaces. Lect. Notes Phys. 622, 245 (2003)

    Article  ADS  Google Scholar 

  4. Julve, J., de Urríes, F.J.: Inner products of resonance solutions in 1-D quantum barriers. J. Phys. A, Math. Theor. 43, 175301 (2010)

    Article  ADS  Google Scholar 

  5. Berggren, T.: Expectation value of an operator in a resonant state. Phys. Lett. B 373, 1 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  6. Civitarese, O., Gadella, M., Betan, R.I.: On the mean value of the energy for resonant states. Nucl. Phys. A 660, 255 (1999)

    Article  ADS  Google Scholar 

  7. de la Madrid, R.: The rigged Hilbert space approach to the Gamow states. J. Math. Phys. 53, 102113 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  8. Siegert, G.: On the Derivation of the Dispersion Formula for Nuclear Reactions. Phys. Rev. A 56, 750 (1939)

    Article  ADS  Google Scholar 

  9. Nussenzveig, H.M.: Causality and Dispersion Relations. Academic Press, New York (1972)

    Google Scholar 

  10. de la Madrid, R.: The analytic continuation of the Lippmann-Schwinger eigenfunctions, and antiunitary symmetries. SIGMA 5, 043 (2009)

    Google Scholar 

  11. Berggren, T.: Resonance state expansions in nuclear physics. Lect. Notes Phys. 325, 105 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  12. García-Calderón, G., Peierls, R.: Resonant states and their uses. Nucl. Phys. A 265, 443 (1976)

    Article  ADS  Google Scholar 

  13. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, vol. 297. Dover, New York (1965)

    Google Scholar 

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Acknowledgements

Work supported by the Spanish MICINN/MINECO project FIS2011-29287 and CAM research consortium project S2009/ESP-1594. J. Julve acknowledges the hospitality of the Dipartimento di Fisica dell’Università di Bologna, Italy, where part of this work was done.

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Appendices

Appendix A: Boundary Conditions and Pole Location

We work out the relationship between the Outgoing BCs, the Incoming BCs, and the location of the momenta in the complex plane. We first re-derive for 1-D quantum systems the known link between the Outgoing BCs and the resonances in the lower half plane [9, 12].

We re-write (1) in the more convenient form

$$ \partial_x^2u(x)= \bigl[2mV(x)-p^2 \bigr] u(x) $$
(29)

and consider the Outgoing BCs (4) together with their complex conjugates. Computing the expression

$$ \int^L_0{\rm d}x \bigl[u^* \partial_x^2u-u \partial_x^2u^* \bigr] $$
(30)

both using (29) and integration by parts, yields the equality

$$ \bigl(p^{*2}-p^2\bigr)\int ^L_0{\rm d}x |u|^2={\rm i} \bigl(p+p^*\bigr) \bigl(|u(0)|^2+|u(L)|^2\bigr) $$
(31)

With \(p=\alpha+{\rm i}\beta\) one obtains

$$ - {\rm i} 4\alpha\beta\int^L_0{ \rm d}x |u|^2={\rm i} 2\alpha \bigl(|u(0)|^2+|u(L)|^2 \bigr), $$
(32)

which, for α≠0, yields

$$ \beta=-\frac{|u(0)|^2+|u(L)|^2}{2\int^L_0{\rm d}x |u|^2}\;\;< 0\;. $$
(33)

The case α=0 (anti-bound solutions) may be dealt with by using the momentum-derivative u′(x)≡ p u(x), which obeys the equations \(\partial_{x}^{2}u'= [2mV-p^{2}] u'-2p u\) and \(\partial_{x}u'|_{ 0}=-{\rm i}p u'(0)-{\rm i} u(0), \partial_{x}u'|_{L}={\rm i} p u'(L)+{\rm i} u(L) \), and starting from the expression

$$ \int^L_0{\rm d}x \bigl[u^* \partial_x^2u'-u' \partial_x^2u^*\bigr]. $$
(34)

Then we obtain the equality

$$\begin{aligned} &- {\rm i} 4\alpha\beta\int^L_0{\rm d}x u^*u' - 2(\alpha+{\rm i}\beta )\int^L_0{ \rm d}x |u|^2 \\ &\quad= {\rm i} 2\alpha \bigl(u^*(0)u'(0)+u^*(L)u'(L) \bigr)+{\rm i} \bigl(|u(0)|^2+|u(L)|^2\bigr) \end{aligned}$$
(35)

which, for α=0, yields the same result (33).

The Incoming BCs lead to β>0 for any α, and give rise to both the incoming resonances and the bound states.

Appendix B: Master Inner Product

The space integral in (10) can be split in three sectors

$$ \langle\mathcal{E}|\mathcal{E'} \rangle= \mathcal{N}^*\mathcal{N}'\left \{ \begin{array} {l} \int^{ 0}_{-\infty}{\rm d}x [\mathcal{D}^*\mathcal{D'}{\rm e}^{-{\rm i}(z^*-z')x}+\mathcal{R}^*\mathcal{R'}{\rm e}^{{\rm i}(z^*-z')x}\\ \quad{}+\mathcal{R}^*\mathcal{D'}{\rm e}^{{\rm i}(z^*+z')x}+\mathcal{D}^*\mathcal{R'}{\rm e}^{-{\rm i}(z^*+z')x}]\\ \quad{}+(\mathcal{N}^*\mathcal{N}')^{-1}\int^L_0{\rm d}x \varPhi^*_\mathcal{E}(x)\varPhi _\mathcal{E'}(x)\\ \quad{}+\int^{\infty}_L{\rm d}x \mathcal{T}^*\mathcal{T'}{\rm e}^{-{\rm i}(z^*-z')x}\\ \end{array} \right . $$
(36)

with the short-hand notation \(\mathcal{N}^{*}\equiv(\mathcal{N}(z))^{*}\), \(\mathcal{N'}\equiv\mathcal{N}(z')\), and similarly for \(\mathcal{D}\) and the amplitudes \(\mathcal{R}\) and \(\mathcal{T}\). We then bring all the infinite-limited integrals to the form \(I(k)=\int_{0}^{\infty}{\rm d}x {\rm e}^{{\rm i}kx}\), so that

$$ \langle\mathcal{E}|\mathcal{E'} \rangle= \mathcal{N}^*\mathcal{N}'\left \{ \begin{array} {l} I(z^*-z')\mathcal{D}^*\mathcal{D'}+I(-z^*+z')(\mathcal{R}^*\mathcal{R'}+\mathcal{T}^*\mathcal{T'})\\ \quad{}+I(-(z^*+z'))\mathcal{R}^*\mathcal{D'}+I(z^*+z')\mathcal{D}^*\mathcal{R'}\\ \quad{}+(\mathcal{N}^*\mathcal{N}')^{-1}\int^L_0{\rm d}x \varPhi^*_\mathcal{E}(x)\varPhi _\mathcal{E'}(x)\\ \quad{} -\mathcal{T}^*\mathcal{T'}\int^L_0{\rm d}x {\rm e}^{-{\rm i}(z^*-z')x}\\ \end{array} \right . $$
(37)

The form (8) of the solution at x=0 and at x=L lets expressing \(\int^{L}_{0}{\rm d}x \varPhi^{*}_{\mathcal{E}}\varPhi_{\mathcal{E'}}\) in terms of the amplitudes \(\mathcal{D}\), \(\mathcal{R}\) and \(\mathcal{T}\) outside the barrier. The procedure uses the operator \(\partial^{2}_{x}\) for which \(\partial^{2}_{x}\varPhi_{\mathcal{E}}=2m(V-\mathcal{E})\varPhi_{\mathcal{E}}\), and integration by parts to obtain

$$\begin{aligned} \bigl(z^{*2}-z'^2\bigr)\int ^L_0{\rm d}x \varPhi^*_\mathcal{E} \varPhi_\mathcal{E'} =&2m\bigl(\mathcal{E}^*- \mathcal{E'}\bigr)\int^L_0{\rm d}x \varPhi^*_\mathcal{E}\varPhi_\mathcal{E'} \\ =&\int^L_0{\rm d}x \varPhi_\mathcal{E}^*(x) \bigl(\overrightarrow{\partial ^2_x}- \overleftarrow{\partial^2_x}\bigr)\varPhi_\mathcal{E'}(x)=W \bigl[\varPhi^*_\mathcal{E}, \varPhi_\mathcal{E'} \bigr]^L_0 \\ =& {\rm i}\bigl(z^*+z'\bigr)\mathcal{N}^* \mathcal{N}'\bigl(-\mathcal{D}^*\mathcal{D'}+ \mathcal{R}^*\mathcal{R'}+\mathcal{T}^*\mathcal{T}'{\rm e}^{-{\rm i}(z^*-z')L}\bigr) \\ &{}+{\rm i}\bigl(z^*-z'\bigr)\mathcal{N}^* \mathcal{N}'\bigl(\mathcal{R}^*\mathcal{D'}- \mathcal{D}^*\mathcal{R}'\bigr) , \end{aligned}$$
(38)

where W[ϕ,ψ]≡ϕ∂ x ψψ∂ x ϕ is the Wronskian, so that

$$\begin{aligned} \int^L_0{\rm d}x \varPhi^*_E\varPhi_{E'} =&\frac{{\rm i} \mathcal{N}^*\mathcal{N}'}{z^*-z'}\bigl(- \mathcal{D}^*\mathcal{D'}+\mathcal{R}^*\mathcal{R'}+ \mathcal{T}^*\mathcal{T}'{\rm e}^{-{\rm i}(z^*-z')L}\bigr) \\ &{}+\frac{{\rm i} \mathcal{N}^*\mathcal{N}'}{z^*+z'}\bigl(\mathcal{R}^*\mathcal{D'}- \mathcal{D}^*\mathcal{R}'\bigr) \end{aligned}$$
(39)

provided that \(\mathcal{E}^{*}\neq\mathcal{E}'\) (i.e. z ∗2z2). The finite last integral term in (37) gives the result \({\rm i}(z^{*}-z')^{-1}\mathcal{T}^{*}\mathcal{T}'(1-{\rm e}^{-{\rm i}(z^{*}-z')L})\), which readily leads to (11).

The case \(\mathcal{E}^{*}=\mathcal{E}'\) arises in the calculation of the norms whether they be finite (bound states) or divergent (all the others). In the latter case, the infinite result shows up already in the integrals I(k) in (37), regardless of the method used to compute the finite integral \(\int^{L}_{0}{\rm d}x \varPhi^{*}_{\mathcal{E}}(x)\varPhi_{\mathcal{E'}}(x)\). We calculate the finite result of the bound states in Sect. 3.3 by using the technique of quadratures.

Appendix C: Infinite-Limited Integrals

For computing I(k) we rely on the limit λ→0 of the basic Gaussian regularized integral

$$ J(k,\lambda)\equiv\int^{+\infty}_0 { \rm d}x {\rm e}^{-\lambda x^2}{\rm e}^{{\rm i}kx }=\frac{{\rm i}}{k} \sqrt{\pi} \tau {\rm e}^{\tau^2} {\rm erfc}(\tau)\quad (\lambda\ {\rm real}>0) $$
(40)

which is directly related to (7.1.2) in [13], where \(\tau=-{\rm i}k/(2\sqrt{\lambda})\), and hence k, can take any complex value. See Appendix A in [4] for more details.

For \(\operatorname{Im}k >0\) the integral is always convergent so that the limit can be taken in the integrand in (40). Then we trivially have \(I(k)\equiv J(k,0)={\rm i} k^{-1}\).

For real k, the quoted result

$$ \int^{\infty}_0{\rm d}x {\rm e}^{{\rm i}kx}= {\rm i} PV\frac {1}{k}+\pi \delta(k) $$
(41)

relies on adding to k a small imaginary part \({\rm i}\epsilon\), which still guarantees the convergence when λ→0, but later in the limit ϵ→0+ the result must be interpreted as a distribution.

For \(\operatorname{Im}k <0\) the integration and the limit λ→0 do not commute and we adopt the limit of the integral as a prescription. From (7.1.23) in [13] we obtain

$$ I(k)\equiv J(k,0)=\left \{ \begin{array}{l@{\quad}l} \frac{{\rm i}}{k},& -\frac{\pi}{4}<{\rm arg}(k)<5\frac{\pi}{4} , \quad k\neq0\\ \infty ,& {\rm otherwise} \end{array} \right . $$
(42)

which extends the finite result \({\rm i}k^{-1}\) to a new region of \(\mathcal{C}_{-}\) and defines the “divergence wedge” there mentioned.

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Julve, J., Turrini, S. & de Urríes, F.J. Inner Products of Energy Eigenstates for a 1-D Quantum Barrier. Int J Theor Phys 53, 971–984 (2014). https://doi.org/10.1007/s10773-013-1890-y

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