Quantum particles in a suddenly moving localized potential
Miguel Ahumada-Centeno Paolo Amore Francisco M Fernández Jesus Manzanares













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    Quantum particles in a suddenly moving localized potential
        
    Miguel Ahumada-Centeno 
    
    Facultad de Ciencias,  Universidad de Colima,
    
    Bernal Díaz del Castillo 340, Colima, Colima, Mexico  
    
    mahumada0@ucol.mx 
    
    Paolo Amore 
    
    Facultad de Ciencias, CUICBAS, Universidad de Colima,
    
    Bernal Díaz del Castillo 340, Colima, Colima, Mexico  
    
    paolo.amore@gmail.com 
    
    Francisco M Fernández 
    
    INIFTA,  Division Quimica Teorica, 
    
    Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina 
    
    fernande@quimica.unlp.edu.ar 
    
    Jesus Manzanares 
    
    Universidad de Sonora, Mexico 
    
    new.jmanza@gmail.com 
    
        Accepted 2019 September 24; received 2019 September 10; in original form 2019 July 12.
    ===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
    
    
      We study the behavior of a quantum particle, trapped in localized potential, when the trapping potential starts suddenly to move with constant velocity. In one dimension we have reproduced the results obtained 
    by Granot and Marchewka, Ref. <cit.>,  for an attractive delta function, using an approach based on  a spectral decomposition, rather than on the propagator. 
    We have also considered the cases of Pöschl-Teller and simple harmonic oscillator potentials (in one dimension) and to the hydrogen atom (in three dimensions). In this last case we have calculated explicitly the leading contribution to the ionization probability for the hydrogen atom due to a sudden movement.
    
    
    § INTRODUCTION
    
    
    In a recent paper, Ref. <cit.>, Granot and Marchewka have studied the interesting problem of determining the behavior
    of  a quantum particle trapped in a localized potential, when the potential suddenly starts to move at constant speed at t=0. These
    authors used an attractive Dirac delta potential in one dimension to model the problem, calculating exactly the probability that
    the particle remains confined to the moving potential or that it remains in the initial position. In addition to these two possibilities, they also 
    observed  the probability that the particle moves at twice the speed of the potential: for an observer sitting in the rest frame
    of the potential at t>0^+ this phenomenon can be interpreted as a quantum reflection of a particle moving to the left with speed -v
    from the well. 
    
    The technological advances are making increasingly realistic the scenarios where a quantum particle, such as an atom, can be 
    trapped by suitable attractive potential (corresponding for example to a tip of a needle in a scanning tunneling microscope or a highly focused laser beam
    in an optical tweezer) and thus be relocated in a different region <cit.>. As remarked by Granot and Marchewka, the quantum nature of this process leads to surprising results, as the possibility that the particle moves at twice the speed of the needle. 
    
    The present paper has two different goals: first, to reproduce the analysis of Ref. <cit.> using a more direct approach based 
    on a spectral decomposition rather than on the propagator; second, to extend the analysis to a  wider class of problems, in one and 
    three dimensions.
    
    The paper is organized as follows: in Section <ref> we discuss the general framework, using spectral decomposition; in 
    Section <ref> we consider several examples of potentials, with spectra which can be either mixed or discrete, and calculate
     explicitly the relevant probabilities for each case; finally in Section <ref> we draw our conclusions.
    
    
    
    § SPECTRAL DECOMPOSITION
    
    
    Our starting point is the time dependent Schrödinger equation (TDSE)
    
        i ħ∂Ψ/∂ t = -ħ^2/2mΔΨ + V(r⃗-v⃗ t) Ψ(r⃗,t)
    
    where V(r⃗-v⃗ t) is a potential moving with velocity v⃗.
    
    Let us define ξ⃗ (r⃗,t) ≡r⃗-v⃗ t and let ϕ(ξ⃗) be the solution of the time independent
    Schrödinger equation (TISE)
    
        - ħ^2/2mΔ_ξϕ  + V(ξ⃗) ϕ(ξ⃗) = E ϕ(ξ⃗)
    
    
    It can be easily verified that 
    
        Ψ(r⃗,t) = e^i m v⃗·r⃗/ħ -i m v⃗^2 t/2ħϕ(ξ⃗) e^-i E t/ħ
    
    is a solution to the time-dependent Schrödinger equation (<ref>).
    
    The physical situation studied by Granot and Marchewka amounts to having a quantum particle at the initial time in the ground state
    of the static potential and determining the probability at later times t>0 that the particle can be found in any of the modes
    of the moving potential. For simplicity we will denote as ψ_n(r⃗,t) and ψ_k⃗(r⃗,t) the eigenmodes of the
    moving potential and ϕ_n(r⃗) and  ϕ_k⃗(r⃗) the eigenmodes of the static potential, where n and k⃗ refer to bound an continuum states, respectively. 
    We also call Φ(r⃗) the initial wave function at t=0 (for the specific case studied in Ref. <cit.> this is 
    the wave function of the ground state).
    
    Let Ψ(r⃗,t) be the wave function solution to the TDSE with a moving potential at t>0, subject to the condition
    Ψ(r⃗,0) = Φ(r⃗); this wave function can be naturally decomposed in the basis of the moving potential as
    
        Ψ(r⃗,t) = ∑_n a_n ψ_n(r⃗,t) + ∫d^3k/(2π)^3  b(k⃗) ψ_k⃗(r⃗,t)
    
    where, using eq. (<ref>),
    
        a_n    =   ∫ d^3r ψ_n^⋆(r⃗,0) Φ(r⃗) = ∫ d^3r e^- i m v⃗·r⃗/ħϕ_n^⋆(r⃗,0) Φ(r⃗)  
        
        b(k⃗)    =   ∫ d^3r ψ_k⃗^⋆(r⃗,0) Φ(r⃗) = ∫ d^3r e^- i m v⃗·r⃗/ħϕ_k⃗^⋆(r⃗,0) Φ(r⃗)   .
    
    
    In this way ∑_n |a_n|^2 is the probability that the particle remains in a bound state when the potential starts moving, whereas
    ∫d^3k/(2π)^3 |b(k⃗)|^2 is the probability that the particle will escape to the continuum [We are discussing here the three dimensional case, but the modifications for the one and two dimensional cases are straightforward.].
    
    Calling N the number of bound states of the potential, we define the physical amplitudes 
    
        𝒬_ij(v⃗)    ≡   ∫_-∞^∞ e^- i m v⃗·r⃗ /ħ[ϕ_j(r⃗)]^⋆ϕ_i(r⃗) d^3 r 
        𝒫_i(k⃗,v⃗)    ≡   ∫_-∞^∞ e^- i m v⃗·r⃗ /ħ[ϕ(k⃗,r⃗)]^⋆ϕ_i(r⃗) d^3r 
        𝒫̃_i(k⃗,v⃗)    ≡   ∫_-∞^∞ e^- i m v⃗·r⃗ /ħϕ_i^⋆(r⃗) ϕ(k⃗,r⃗) d^3r = 𝒫̃_i^⋆(k⃗,-v⃗)
    
    
    The wave function at later times for a particle initially in the ith state of the static potential will then be
    
        Ψ(r⃗,t) = ∑_j=1^N 𝒬_ijψ_j(r⃗,t) +  ∫_-∞^∞𝒫_i(k⃗,v) ψ(k⃗,r⃗,t) d^3k/(2π)^3
    
    
    Clearly |𝒬_ij|^2 represents the probability that the particle, initially in the ith bound state ends up in the
    jth bound state; similarly |𝒫_i(k⃗,v)|^2 d^3 k/(2π)^3 represents the probability that the particle initially in the ith bound state ends up in the continuum with momentum in an infinitesimal volume about ħk⃗.
    
    The conservation of total probability requires
    
        ∑_j=1^N |𝒬_ij|^2 + ∫_-∞^∞ |𝒫_i(k⃗,v)|^2 d^3k/(2π)^3 = 1
    
    
    Similarly we define the amplitude
    
        ℛ(k⃗,k⃗',v⃗)    ≡   ∫_-∞^∞ e^- i m v⃗·r⃗ /ħ[ϕ(k⃗',r⃗)]^⋆ϕ(k⃗,r⃗) d^3r
    
    which is related to the probability that a particle initially with momentum ħk⃗ ends up in a state with momentum ħk⃗'.
    
    Of course these arguments can be generalized to the case that the initial wave function is not a stationary state of
    the static problem. In this case the time dependent wave function is 
    
        Ψ(r⃗,t) = ∑_j=1^N 𝒬̅_jψ_j(r⃗,t) +  ∫_-∞^∞𝒫̅(k⃗,v) ψ(k⃗,r⃗,t) d^3k/(2π)^3
    
    where
    
        𝒬̅_j    ≡   ∫ e^- i m v⃗·r⃗/ħ[ϕ_j(r⃗)]^⋆Φ(r⃗) d^3r 
           =   ∑_i=1^N a_i 𝒬_ij +  ∫ b(k⃗) 𝒫̃_j(k⃗,v⃗) d^3k/(2π)^3
        𝒫̅(k',v)    ≡   ∫ e^- i m v⃗·r⃗ /ħ[ϕ(k⃗',r⃗)]^⋆Φ(r⃗) d^3r
    
    of which Eq.(<ref>) is a special case.
    
    
    
    
    § APPLICATIONS
    
    
    In this section we apply our general discussion to different examples.
    
    
    
     §.§ Attractive Dirac delta potential
    
    
    We consider the attractive Dirac delta potential studied in Ref. <cit.>:
    
        V(x,t)={[    -γδ(x)         ,     t ≤ 0; -γδ(x-vt)         ,     t > 0;           ].
    
    
    To implement the procedure explained in Section <ref> we first write explicitly 
    the eigenfunctions of the static potential, V(x,0)= -γδ(x), reported in Ref. <cit.>
        ϕ_0(x)    =   √(β) e^-β| x| 
        ϕ_p^(e)(x)    =   √(2) (p cos (p x)-βsin (p | x| ))/√(β ^2+p^2)
        ϕ_p^(o)(x)    =   √(2)sin (p x)
    
    where β = m γ/ħ^2. Damert <cit.> and Patil <cit.> have proved the completeness 
    of the set of the energy eigenfunctions.
    Note that the spectrum of V(x,0) is mixed with a single bound state of energy E_0 = - γ ^2 m/2 ħ ^2.
    
    The orthonormality relations for this set of functions are
    
        ∫_-∞^∞ϕ_0^⋆(x) ϕ_0(x) dx    =    1 
        ∫_-∞^∞ϕ_0^⋆(x)  ϕ_p^(e)(x) dx    =   ∫_-∞^∞ϕ_0^⋆(x)  ϕ_p^(o)(x) dx = 0  
        ∫_-∞^∞ϕ_p'^(e)^⋆(x)  ϕ_p^(e)(x) dx    =   ∫_-∞^∞ϕ_p'^(o)^⋆(x)  ϕ_p^(o)(x) dx =  2 πδ (p-p')  
        ∫_-∞^∞ϕ_p'^(o)^⋆(x)  ϕ_p^(e)(x) dx    =    0
    
    
    Using the prescription (<ref>) one can obtain the solutions to the time dependent problem for t>0 corresponding
    to the static solutions of eqs. (<ref>), (<ref>) and (<ref>):
    
        ψ_0(x,t,v)    =   √(β) e^-β| x-t v| +i m (γ ^2 t+v ħ ^2 (2 x-t v))/2 ħ^3
        ψ_k^(e)(x,t,v)    =   √(2)/√(β ^2+k^2)  e^-i (k^2 t ħ ^2+m^2 v (t v-2 x))/2 m ħ 
        (k cos (k (x-t v))-βsin (k | x-t v|))  
        ψ_k^(o)(x,t,v)    =   √(2)sin (k (x-t v)) e^-i (k^2 t ħ ^2+m^2 v (t v-2 x))/2 m ħ
    
    
    The initial wave function is the bound state of the static delta potential
    
        Ψ_0(x)    =   √(β) e^-β| x|
    
    and it can be decomposed in the basis of the time-dependent potential as
    
        Ψ_0(x) = 𝒬_11(v) ψ_0(x,0) + ∫_0^∞dk/2π[𝒫_1^(e)(k,v) ψ_k^(e)(x,0)+
        𝒫_1^(o)(k,v) ψ_k^(o)(x,0)]
    
    as explained in Section <ref>.
    
    The amplitude for transition to the bound state of the moving well is given by
    
        𝒬_11(v)    =   ∫_-∞^∞ e^- i m v x/ħϕ_0^⋆(x) ϕ_0(x) dx  = 4/θ ^2+4
    
    where θ≡ħ v/γ is the adiabatic Massey parameter <cit.>. 
    
    Clearly, the probability that the particle remains in the bound state of the moving well is simply given by
    
        P_bound= |𝒬_11|^2 = 16/(θ^2+4)^2
    
    in agreement with the eq.(13) of Ref. <cit.>.
    
    Similarly we can calculate the coefficients 𝒫_1^(e)(k,v) and 𝒫_1^(o)(k,v), representing the amplitudes for transitions to a state in the continuum (even and odd states, respectively)
    with momentum ħ k
        𝒫_1^(e)(k,v) 
           =   ∫_-∞^∞  e^- i m v x/ħ[ϕ_k^(e)(x)]^⋆ϕ_0(x) dx 
           =   4 √(2)β^7/2θ ^2 k/√(β^2+k^2)(β^4 (θ ^2+1)^2+k^4-2 β ^2 (θ ^2-1) k^2)
        𝒫_1^(o)(k,v)
           =   ∫_-∞^∞  e^- i m v x/ħ[ϕ_k^(o)(x)]^⋆ϕ_0(x) dx 
           =    -4 i √(2)β ^5/2θ k/β ^4 (θ^2+1)^2+k^4-2 β ^2 (θ^2-1) k^2
    
    where β  = m v/ħθ.
    
    As a result the probability that the particle ends up in the continuum is
    
        P_continuum   =   ∫_0^∞[ |𝒫_1^(e)(k,v)|^2 + |𝒫_1^(o)(k,v)|^2] dk/2π
           =   ∫_-∞^∞[ |𝒫_1^(e)(k,v)|^2 + |𝒫_1^(o)(k,v)|^2] dk/4π
        
           =   ∫_-∞^∞16 β ^5 θ ^2 k^2 (β ^2
        	(θ^2+1)+k^2)/(β^2+k^2) (β ^4 (θ^2+1)^2+k^4-2 β ^2 (θ^2-1) k^2)^2dk/2π
    
    
    A straightforward integration of this expression using the residue theorem provides the final result
    
        P_continuum   =    1-16/(θ ^2+4)^2
    
    
    The total probability correctly sums to 1:
    
        P_total = P_bound+P_continuum = 1
    
    
    The solution at t >0 is then obtained using eq. (<ref>):
    
    
    
    using this expression we were able to reproduce Fig. 3 of Ref. <cit.> performing a numerical integration 
    of this equation (notice however a typo in the second equation of (15) of Ref. <cit.>).
    
    
    
     §.§ Pöschl-Teller potentials
    
    
    The second example that we want to consider is the Pöschl-Teller (PT) potential
    
        V(x) = -ħ^2 λ  (λ +1)/2 a^2 m ^2(x/a)
    
    for which exact solutions are available.
    
    The potential (<ref>) provides a nice generalization of our discussion for the attractive delta potential, both because
    it has a mixed spectrum, with λ bound states (λ integer), and because it is known to be reflectionless <cit.>.
    
    The eigenfunctions of the  bound states read
    
        ϕ_j(x) = 𝒩_j/√(a) P_λ^j(tanh(x/a))   ,   j=1, …, λ
    
    where P_λ^j(x) is the associated Legendre polynomial; the corresponding eigenenergies are
    E_j= -j^2 ħ ^2/2 a^2 m. Here 𝒩_j is a (dimensionless) normalization constant.
    
    As an example we consider the case λ=1, for which
    
        ϕ_1(x)    =    -√(1-tanh^2(x/a))/√(2)√(a)
        ϕ_k(x)    =   (-tanh(x/a)+i a k)/1+i a k e^i k x
    
    and assume that the particle is in the ground state of the static potential at t=0.
    
    The amplitudes for the transition to the bound state and to the continuum states of the moving potential can then be
    calculated explicitly as
    
        𝒬_11   ≡   ∫_-∞^∞ e^- i m v x /ħ[ϕ_1(x)]^⋆ϕ_1(x) dx 
        = 1/2πκ(πκ/2) 
        𝒫_1(k,v)    ≡   ∫_-∞^∞ e^- i m v x /ħ[ϕ(k,x)]^⋆ϕ_1(x) dx 
        = π√(a)κ/√(2) (a k+i) (1/2π  (a k+κ ))
    
    where κ≡ a mv/ħ.
    
    
    
    In the left plot of Figure <ref> we plot the probability that a particle initially in the bound state of the Poschl-Teller potential with  λ=1 stays trapped as the well starts to move with constant velocity (blue curve),
    P_bound(κ) = | 𝒬_11 |^2, and the probability that the particle ends up in the continuum (red curve),
    P_continuum(κ) = ∫_-∞^∞ | 𝒫_1(k,v) |^2 dk/2π. The integral over momentum
    is performed numerically and it is verified within the numerical accuracy that P_bound(κ)+ P_continuum(κ) =1.
    The situation is qualitatively similar to the case treated in Ref. <cit.>, but with P_bound(κ) decaying now exponentially 
    for κ≫ 1.
    
    The wave function at t>0 can be then obtained as
    
        Ψ(x,t)    = 1/2πκ(πκ/2) ψ_1(x,t) 
           +  ∫_-∞^∞π√(a)κ/√(2) (a k+i) (1/2π  (a k+κ ))  ψ(k,x,t) dk/2π
    
    where
    
        ψ_1(x,t)    =    e^i m v x/ħ -i m v^2 t/2ħϕ_1(x-vt) e^-i E_1 t/ħ
        ψ(k,x,t)    =    e^i m v x/ħ -i m v^2 t/2ħϕ(k,x-vt) e^-i ħ k^2 t/2m
    
    
    Since the Pöschl-Teller potential is reflectionless we expect that the wave function would be qualitatively different from the wave function of the moving delta well, due to the absence of a peak moving with velocity 2v.
    The time evolution of a wave function for a particle initially in the bound state of the PT potential, which suddenly starts to move
    with constant velocity, is displayed in the right plot Fig. <ref>. We have used a=m=ħ=κ=1; the probability density is plotted at four different times t=0,5,10,15. In this case it is evident the absence of the reflected wave, as expected, given the
    nature of the potential. We also appreciate that the peak is moving at velocity v, and the corresponding 
    wave function is not dispersing.
    
    
    
    
    
    The plots in Figs. <ref>  are the analogous of the plots in Figs. <ref>, but for a potential with λ=2;  
    in this case the potential possess two bound states and, as the potential starts to move, the probability of exciting 
    the first excited state of the well grows up to a maximum (for κ≈ 1.5) and then decreases (see the left plot in Fig. <ref>). 
    The conservation of total probability is verified numerically to hold.
    
    
    The location of the maximum of the probability of exciting a different bound state approximately corresponds to absorbing 
    the kinetic energy of the particle and make a transition to the other bound state
    
        - ħ^2/2 m a^2( μ^2 - λ^2 ) = 1/2 m v^2
    
    or equivalently
    
        κ = √(λ^2 -μ^2)
    
    For the case in the left plot of Fig. <ref>, λ = 2 and μ=1, and therefore the maximum corresponds to 
    κ = √(3).
    
    In Fig. <ref> we display the time evolution of the wave function for λ=2 but for κ=2, at which the components of
    the bound states are comparable: in this case one can appreciate the asymmetric time-dependent shape of the peak, which reflects the fact that the  particle is not in a stationary state.
    
    
    
     §.§ Simple harmonic oscillator
    
    
    The simple harmonic oscillator 
    
        V(x) = 1/2 m ω^2 x^2
    
    is possibly the most important example of quantum mechanical problem for which exact solutions are known.
    
    In this case the spectrum is discrete, with bound states of energy
    
        E_n = ħω(n+1/2)    ,    n=0,1,…
    
    and with eigenfunctions 
    
        ψ_n(x) = 1/√(2^n n!)(m ω/πħ)^1/4 e^-m x^2 ω/2 ħ
        H_n(x √(m ω/ħ))
    
    where H_n(x) is the Hermite polynomial of order n.
    
    
    We define the dimensionless parameter 
    
        κ≡m v^2/2ħω
    
    representing  the ratio between the kinetic energy associated with the motion of the well and a quanta of energy ħω.
    
    Assuming that the particle is initially in the ground state of the static potential, the amplitudes can be obtained explicitly;
    the first few amplitudes are
    
        𝒬_0,0(κ) = e^-κ /2      ,      𝒬_0,1(κ) = -i e^-κ /2√(κ)
        𝒬_0,2(κ) = -e^-κ /2κ/√(2)      ,      𝒬_0,3(κ) = i e^-κ /2κ^3/2/√(6)
    
    
    In this case the maximum of the probability | 𝒬_0n|^2 of exciting the state n 
    corresponds a velocity given by
    
        m v^2/2 = ħω n
    
    or equivalently
    
        κ=n
    
    
    The probabilities of transition from the ground state of the SHO to an excited state, due to a sudden movement of the
    well, are plotted in Fig. <ref>. The vertical lines in the plot correspond to the condition (<ref>).
    It is worth noticing that at κ = n, |𝒬_0,n(n)|^2 = |𝒬_0,n-1(n)|^2, for n>1. The proof of this property can be found in  Appendix <ref>.
    
    
    
    
     §.§ Hydrogen atom
    
    
    Let us now consider a hydrogen atom, initially at rest in the ground state, which is suddenly kicked and the proton 
    starts to move with constant velocity v⃗.
    
    The wave function of the bound states of the (static) hydrogen atom read
    
        ψ_nlm(r,θ,ϕ)    =    R_nl(r) Y_l^m(θ ,ϕ ) 
           =     2^l+1 e^-r/a_0 n√((-l+n-1)!/a_0^3 n^4 (l+n)!)(r/a_0 n)^l L_-l+n-1^2 l+1(2 r/n a_0) Y_l^m(θ ,ϕ )
    
    with n ≥ 1, 0 ≤ l ≤ n-1 and |m| ≤ l.
    
    Based on our general discussion, the amplitude for the transition from the ground state to any excited state reads [Since the
    	quantum number for the third component of angular momentum needs to vanish,m=0, we express the amplitudes as functions of n and l alone.]
        𝒬_1,0;n,l   =   ∫ e^-i μ v r cosθ/ħψ^⋆_n,l,0(r⃗) ψ_1,0,0(r⃗) d^3r
    
    where μ here is the mass of the electron.
    
    Using the partial wave decomposition of a plane wave
    
        e^i μ v r  cosθ/ħ = ∑_l=0^∞ (2l+1)  i^l  j_l(μ v r/ħ)  P_l(cosθ)
    
    one can reduce the expressions to a one dimensional integral
    
        𝒬_1,0;n,l(κ)    =   √(2l+1) i^l ∫_0^∞ j_l(μ v r/ħ) R_10(r) R_nl(r) r^2 dr
    
    
    We have calculated explicitly the amplitudes for 1 ≤ n ≤ 10; their expressions 
    (not reported here) depend uniquely on the dimensionless parameter
    κ≡ħ v/e^2/4πϵ_0 =  μ v a_0/ħ (a_0 is the Bohr radius).
    Just to get an idea, κ = 1 corresponds to a speed v ≈ 0.007892 c; using the root mean square speed for
    hydrogen gas v_rms = √(3 k_B T/M) we can associate a temperature T ≈ 2.2 × 10^8  K. To obtain a sizeable 
    ionization effect on the atoms of a gas trapped in a container by means of the elastic collisions of the individual atoms  with the walls of the container, one should reach incredibly high temperatures [Of course, this argument is only qualitative since the boundary conditions for that problem would be different and the wave functions for the moving and static systems would not be related by eq. (<ref>).].
    
    
    Using these expressions we can calculate exacly the probability of a transition (1,0) → (n,l), with 
    n ≤ N, due to a sudden movement with velocity v⃗ [Notice that for the hydrogen atom there is an
    	infinite number of bound states.]
        P_n ≤ N = ∑_n=0^N ∑_l=0^n-1| 𝒬_1,0;n,l(v) |^2
    
    
    Of course the probability of not ionizing the atom corresponds to using N→∞ in the expression above.
    
    At small velocities (κ≪ 1) we may obtain the leading behavior of this probability
    
        P_n ≤ 6   ≈    1 -0.302617 κ^2-0.576334 κ ^4+…
        
        P_n ≤ 7   ≈    1 -0.297702 κ^2-0.572154 κ ^4 +…
        
        P_n ≤ 8   ≈    1 -0.294468 κ^2-0.569285 κ ^4 +…
        
        P_n ≤ 9   ≈    1 -0.292225 κ^2-0.567241 κ ^4 +…
        
        P_n ≤ 10   ≈    1 -0.290603 κ^2 - 0.565735 κ ^4 +…
    
    
    The coefficients of the contribution of order κ^2 form a nice monotonic sequence, so that one can use extrapolation
    to estimate its value for N →∞ accurately. Moreover, since these coefficients receive contributions only from 
    the transition (1,0) → (n,1), a larger number of coefficients can be calculated with limited effort.
    
    We have used Richardson extrapolation on the sequence of the first 20 coefficients obtaining 
    
        - 0.   28341221595516952094 - 0.78146725925265723860  1/N^2
           + 0.78146725925251190251  1/N^3 +…
    
    showing that the ionization probability for the hydrogen atom goes as
    0.28341221595516952094  κ^2 for κ→ 0.
    
    This result can be confirmed by using the wave functions of the continuum, Ψ_klm(r⃗).
    In this case
    
        𝒫(k)    =   ∫ e^-i μ v r cosθ/ħΨ^⋆_klm(r⃗) ψ_1,0,0(r⃗) d^3r
    
    is the amplitude for the transition from the ground state to the continuum (i.e. its modulus square is the probability of ionizing the atom).
    
    The direct calculation of the ionization probability requires using the continuum wave functions. These wave functions
    are reported for instance in Ref. <cit.> and read:
    
        Ψ_klm(r,θ,ϕ)    =    Y_lm(θ,ϕ)  ϕ_kl(r)/r≡ Y_lm(θ,ϕ)  R_l(k,r)
    
    where (note that the radial wave functions R_l(k,r) obey the k/(2 π) normalization)
    
        ϕ_kl(r)    = 2^l+1  (k r)^l+1/a_0 r (2 l+1)! e^π/2 a_0 k-i k r| Γ(l-i/k a_0+1)| 
           × _1F_1(l+i/k a_0+1;2 (l+1);2 i k r)
    
    
    Using these wave functions we obtain the exact expression for the leading  behavior of the ionization probability as κ→ 0
        ∫_0^∞   | 𝒫(k) |^2 dk/2π≈
           κ^2  ∫_0^∞256 π  u
        	(u+i/-u+i)^-i/u(-1+2 i/u+i)^i/u((π/u)+1)/3 (u^2+1)^5du/2π + O(κ^4) 
           ≈ 0.28341221595516952089  κ^2
    
    
    The value obtained using the extrapolation of the complementary probabilities provides a very accurate estimate 
    (with an error approximately of 4.7 × 10^-20). 
    
    
    
    § CONCLUSIONS
    
    We have extended the one dimensional model of Granot and Marchewka in Ref. <cit.> 
    for an atom displacing with a moving tip (represented by a Dirac delta function) to a number of
    potentials with different spectrum (both discrete and mixed) and in one and three dimensions.
    Our calculations are based on a spectral decomposition, rather than on the direct use of the 
    propagator (as done in Ref. <cit.>) and, for the case discussed in <cit.> 
    we reproduce the probability that the particle stays trapped calculated by Granot and Marchewka.
    
    The remaining examples that we discuss present new and interesting features, not found in the example 
    considered in Ref. <cit.>: for instance, for the case of Pöschl-Teller potentials 
    we show that the reflected peak moving with velocity 2v  found in Ref. <cit.>  is absent, 
    due to the reflectionless nature of PT potentials; moreover, for the case of potentials with more 
    than one bound state, there is a probability that the particle gets to an excited bound state, rather 
    than to the continuum and we have found a simple criterium based on energy conservation to identify 
    the maxima of this probability. Finally, we have calculated  exactly  the leading contribution
    in the velocity v to the probability that a hydrogen atom gets ionized due to a sudden movement 
    of the proton.
    
    
    
    
    
    § AMPLITUDES FOR THE SIMPLE HARMONIC OSCILLATOR
    
    
    
    We can understand this result in terms of the creation and annihilation operators 
    â and â^†; calling |n⟩ an eigenstate of the Hamiltonian 
    of the simple harmonic oscillator we have
    
        â^†â|n⟩   =    n|n⟩
        â|n⟩   =   √(n)|n-1⟩
        â^†|n⟩   =   √(n+1)|n+1⟩.
    
    
    Following Ref. <cit.> we define 
    
        I_mn≡⟨ m | U | n ⟩
    
    where Û = e^-αx̂ and x̂ = √(ħ/2mω)(â + â^†).
    
    Then
    
        I_0,n    = ⟨ 0|Û|n⟩ = ⟨ 0|Ûâ^†/√(n)|n-1⟩ = 1/√(n)⟨ 0|(â^†-α√(ħ/2mω))Û|n-1⟩
            = -α/√(n)√(ħ/2mω)⟨ 0|Û|n-1⟩ = -α/√(n)√(ħ/2mω) I_0,n-1.
    
    
    On the other hand I_0,0 = e^α^2 ħ/4 mω and for κ/2 = -ħ/4mωα^2 we obtain 
    
        I_0,n(κ) = -i√(κ/n)I_0,n-1(κ)
    
    from which the property
    
        |I_0,n(n)|^2 = |I_0,n-1(n)|^2
    
    follows.
    
    
    
    
    § ACKNOWLEDGEMENTS
    
    The research of P.A. was supported by Sistema Nacional de Investigadores (México). 
    
    
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