On quantum bound states in equiperiodic multi-well potentials I: − Locally periodic potential sequences and Floquet/Bloch bands
OOn quantum bound states in equiperiodic multi-well potentials I: − Locally periodic potential sequences and Floquet/Bloch bands.
Karl-Erik Thylwe
Kl¨overv¨agen 16, 387 36 Borgholm, Sweden(Retired from Department of Mechanics,KTH- Royal Institute of Technology)
Two connected equiperiodic one-dimensional multi-well potentials of different well depths arestudied. Floquet/Bloch energy bands for respective multi-well potential are found to be relevant forunderstanding level structures. Althoug energies are classically allowed in both multi-well potentials,a band gap of one multi-well potential makes this potential quantum-mechanically ’forbidden’. Allenergy levels are located in the union of the band regions.
I. INTRODUCTION
Floquet/Bloch-type problems in one dimension can be studied by the amplitude-phase method [1]. The methodallows calculations of bound states by a Bohr-Sommerfeld-type quantization condition even for multi-well potentials[2]. Single truncated periodic potentials surrounded by vanishing potentials on both sides has also been analysed forscattering energies [3]. Floquet characteristics such as energy bands and energy gaps are found to be valuable analyticconcepts for explaining energy bands of total transmission [4].Bound-state problems for double-well potentials are treated with various amplitude-phase method without the useof Floquet theory [2]. The doubling of energy levels of such potentials is well known, with larger level splittings formore massive barriers between the wells. Generalization to multi-well potentials is simplified if the wells are identical(’truncated periodic potentials’), which allows identification of band/gap energy regions. A further generalizationconsidered, is the combination of two such multi-well potentials attached to each other. Both multi-well potentialsare assumed having the same periodicity.Recent research on multi-well or multi-barrier potentials [5–8] often relates to physical and technical propertiesof linear nano structures. It focuses on transmission behaviors of external (charged) particles. The formalism formulti-well potentials is similar to that for multi-barrier potentials [9–15]. Total transmission was discussed in energytransmission bands known to contain at most N − N -barrier/well potentials [16].This assertion is modified in [3], where the number of peaks in a transmission band is shown to be either N or N − A ( x ) cos p ( x ) and A ( x ) sin p ( x ), where an amplitude function A ( x ) and a phase function p ( x )are defined along an x -axis in space. The amplitude functions satisfy a Milne-Penny equation [17–19] and phases aredefined by these amplitude functions.Section II presents the second-order Schr¨odinger equation in non-dimentional form. In section III the general Bohr-Sommerfeld condition is derived and energy levels are calculated. Section IV discusses the relevant Floquet/Blochbands involved. Conclusions are in section V. II. BOUND-STATES
The time-independent Schr¨odinger equation with a dimensionless space coordinate x is given by [1] F (cid:48)(cid:48) + 2 m ( E − V ( x )) F = 0 , (1)where a prime ( (cid:48) ) means differentiation with respect to x . The dimensionless symbol m represents an effective massand equation (1) is expressed as if being in atomic units. V ( x ) represents a potential energy function that vanishesoutside an interval 0 ≤ x ≤ N π , where N is the number of potential periods. E ( <
0) represents the total energy.Equation (1), with V ( x ) being a periodic function of x , is a special case of a so-called ’Hill equation’ [20, 21] .Two truncated periodic potentials are connected with π being the unit length of a period cell. The total numberof such cells is N , an even number, with N/ V ( x ) = v sin ( x ) , ≤ x ≤ N π/ , V ( x ) = v sin ( x ) , N π/ ≤ x ≤ N π, (2) a r X i v : . [ qu a n t - ph ] J u l − − − − x / (cid:47) V ( x ) FIG. 1: Illustration of multi-well potential type considered. The total number of potential cells in the figure is N = 6. Potentialparameters are v = − .
25 and v = − . where v is varied and v is fixed here. The potential vanishes outside the x -region in (2).Bound-state solutions F ( x ) satisfy F ( x ) → , | x | → ∞ . (3)An approach using the separate intervals −∞ < x ≤ N π/
N π/ ≤ x < ∞ is presented. Solutions satisfying(3) are integrated from x = N π/
III. GENERAL BOHR-SOMMERFELD QUANTIZATION CONDITION FOR BOUND STATES
Amplitude functions satisfy a non-linear differential equation, the Milne-Pinney equation [17]-[19]. Any relevantwave function is defined by them.Two independent wave solutions of (1) are defined in terms of a positive amplitude function A ( x ) and a relatedreal phase function p ( x ) as [1] Ψ ( ± ) ( x ) = A ( x ) exp( ± i p ( x )) , (4) p (cid:48) ( x ) = A − ( x ) ( > . (5)Relation (5) makes sure that the Wronskian of the two solutions (4) is independent of x , see reference [22]. Anamplitude function satisfies a nonlinear Milne-Pinney equation A (cid:48)(cid:48) ( x ) + 2 [ E − V ( x )] A ( x ) = A − ( x ) . (6)Amplitude functions differ by their boundary conditions [23] and are more or less oscillating. For molecular massesthe choice of slowly varying amplitude functions are typically inspired by the WKB amplitude functions, which areslowly varying in each classically allowed region.Equation (6) is integrated as a first-order differential equation A ( x ) A (cid:48) ( x ) p ( x ) (cid:48) = A (cid:48) ( x ) A − ( x ) − E − V ( x )) A ( x ) A − ( x ) . (7)An integration starts at a boundary point with boundary conditions for the amplitude function. The phase functionneeds an additional integration constant to be specified. These equations can be applied in various ways. Often theequations are applied several times. For example in each characteristic region of the potential; i.e. in locally periodicregions and in exterior, asymptotic regions.The most simple bound-state representation of the Schr¨odinger wave is F ( x ) = A ( x ) sin φ ( x ) , (8)where the phase function φ ( x ) includes a specific integration constant, while p ( x ) in (7) has an unspecified integrationconstant. For bound states the representation (8) tends to zero as | x | → ∞ . Since A ( x ) (cid:54) = 0 one requires sin φ ( x ) → − − − − − − − − − − − − − − − − − − − − E − − − − − − − − − − N = 2 N = 4 N = 6 FIG. 2: Levels corresponding to the two potential cases ( v , v ) = ( − . , .
25) (solid lines) and ( v , v ) = ( − . , .
25) (dashedlines). The broader solid (black) and dashed (red) lines indicate edges of the two Floquet/Bloch bands involved. j ( v , v ) = ( − . , − . E j ( v , v ) = ( − . , − . E j N = 2Band edge E = − E = − E = − E = − N = 2 corresponding to Figure 1. The case ( v , v ) = ( − . , − .
25) corresponds to asymmetric double-well potential. The case ( v , v ) = ( − . , − .
25) corresponds to an asymmetric double-well potential. as x → −∞ , and also sin φ ( x ) → x → ∞ . In fact, A ( x ) → ∞ as | x | → ∞ . However, it can be shown thatsin φ ( x ) → A ( x ) diverges in these x -limits. Boundary conditions for the quantities in (7) are chosen as A ( N π/
2) = 1 , A (cid:48) ( N π/
2) = 0 , p ( N π/
2) = 0 . (9)The position x = N π/ x = N π/ x = − x L and at some sufficiently largepositive x = x R . Two phase values are obtained after the integrations: α = − p ( − x L ) and β = p ( x R ). The relevantphase function φ ( x ) in (8) is defined as φ ( x ) = p ( x ) + α , and its phase value as x → + ∞ is φ (+ ∞ ) = α + β . Thebound state condition is α + β = ( j + 1) π, j = 0 , , , · · · . (10)In this way the wave function F ( x ) is normalized to satisfy F ( N π/
2) = sin α and F (cid:48) ( N π/
2) = cos α .The method resulting in (10) is exact. It is numerically best suitable for light-particle interaction potentials witheffective masses m satisfying (0) < m <
5, and cell numbers satisfying (0) ≤ N <<
10. The time efficiency isgradually lost for larger values of m and N .Figure 2 shows energy levels for two multi-well combinations, ( v , v ) = ( − . , .
25) (solid lines) and ( v , v ) =( − . , .
25) (dashed lines), and N/ v respectively v ). Hence, the solid level linescorrespond to single-type N -well potentials; N = 2 corresponding to an ordinary double-well potential, N = 4 to anordinary quadruple-well potential, and so forth. As N increases, the levels become more and more dense. However,the levels are contained in a finite range of energies, a particular Floquet/Bloch band region corresponding to v or v = − . v , v ) = ( − . , . N increases. They appear in two groups, contained in the two relevant Floquet/Bloch bands, one band j ( v , v ) = ( − . , − . E j ( v , v ) = ( − . , − . E j N = 4Band edge E = − E = − E = − E = − N = 4 corresponding to Figure 1. The case ( v , v ) = ( − . , − .
25) corresponds to asymmetric quadruple-well potential. The case ( v , v ) = ( − . , − .
25) corresponds to an asymmetric quadruple-well potential. corresponding to v , and one band corresponding to v . These energy bands do not overlap. Energy levels are notobserved between the band regions. IV. FLOQUET/BLOCH BANDS
Any periodic potential with a given value of the strength has specific energy Floquet/Bloch bands. Each value of v and/or v is associated with several, more or less separated, energy bands. For single-well potentials correspondingto either v or v , one has two sets of bound-state levels. Each level in each potential develops into a Floquet/Blochband, as more and more identical wells are added. The phenomenon is a kind of discrete multi-well level splitting,approaching a continuous energy band as the number of identical wells tends to infinity.The resulting bands corresponding to v and v may, or may not, overlap. Only the bands corresponding to groundstate levels of the single wells are considered here.Energy band edges satisfy any of the conditions [1] u cos γ = ± . (11)The symbol u represents an amplitude value after integrating (7) accross one cell (a single well) of the potential.Boundary conditions for the amplitude function are A ( x b ) = 1, A (cid:48) ( x b ) = 0 and p ( x b ) = 0. The position ’ x b ’ is chosenas either ’0’, for the left multi-well potential in Figure 2, or ’ N π/ γ represents the value of p ( x ) in (7) obtained ofter integration at x = x b + π . The band edges satisfy u cos γ = 1or u cos γ = − between the relevant bands. V. CONCLUDING REMARKS
The multi-well potentials in this study are locally periodic and can be associate with band/gap properties. Onefinds that a multi-well potential attached to another multi-well potential sees its neigbor potential as a ’forbidden’region at certain energies, even if the energy is classically allowed in both potentials. The energy then happens to liein a band gap of the neighbor potential. A consequence is a small shift of energy levels compared to the unconnectedmulti-level potential. As the multi-well potentials become identical, all levels are confined to a single Floquet/Blochenergy band.A feature of the amplitude-phase approach for quantal waves is that amplitude functions can be defined by differentboundary conditions without changing the original quantal wave. Other approaches may explore this ’amplitudefreedom’ to treat characteristic regions (exterior or locally periodic ones) by completely separate amplitude-phase j ( v , v ) = ( − . , − . E j ( v , v ) = ( − . , − . E j N = 6Band edge E = − E = − E = − E = − N = 4 corresponding to Figure 1. The case ( v , v ) = ( − . , − .
25) correspondsto a symmetric quadruple-well potential. The case ( v , v ) = ( − . , − .
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