A.1. Bound states of an electron on the surface of a dielectric liquid The wavefunctionψ(x, y, z)of an electron on the surface of a dielectric liquid is the solution of Schrödinger equation
−h2
2m∇2ψ(x, y, z) −Λ
zψ(x, y, z) =Eψ(x, y, z)=⇒ (A.1)
∇2ψ+
2mΛ h2z +2mE
h2
ψ=0, (A.2)
where it is reminded thatΛhas been defined in eq. (1.1) andEdenotes the eigenenergies of the bound states, meaning thatE < 0; for negative eigenenergies, it can be demanded that the wavefunction will be zero at the origin(z=0)and at infinity(z−→∞).
The laplacian is separable in cartesian coordinates, and the substitution ψ(x, y, z) =f(x, y)φ(z)into eq. (A.1) leads to a wave equation for(x, y). Decom-posing the total eigenenergy into the sumE=Ek+Eν, one gets
∇2f(x, y) =2mEk
h2 f(x, y). (A.3)
This equation can be further separated into two one-dimensional wave equa-tions whose soluequa-tions are plane waveseikxxandeikyy(here, the wavenumbers kx,y=h2k2/2m,fork=1, 2, . . .were introduced, such thatEk=h2k2/2m).
Next, we define the “natural dimensions” of the problem, by setting a=2h2
mΛ≈78Å, λ= r
−2mE
h2 , ν= 1
λa, ζ=λz. (A.4) Observe that the parametera(the “Bohr radius” of the problem) has dimen-sions of length, whileλhas dimensions of inverse length, making the productλa dimensionless. Moreover, the new argumentζofφ(·)is dimensionless as well.
These transformations, lead to a dimensionless form of the differential equation forφ(ζ), which is suitable for further mathematical manipulations. The dimensionless form is
φ00(ζ) + 1
λaζ−1 4
φ(ζ) =0=⇒φ00(ζ) + ν
ζ−1 4
φ(ζ) =0. (A.5)
137
Figure A.1. Probability den-sities|φν(z)|2of bound states νfor a single electron above the surface of liquid4He, as a function of the normalized distancez/a, wherea=78Å (cf. eq. A.9). The average electron of the ground state is found approximately 78Å above the surface of liquid he-lium.
The previous ordinary differential equation is not yet recognized as belong-ing to one of the known solvable forms, but by means of the transformation
φ(ζ) =ζe−ζ/2u(ζ), (A.6) it is recasted into the Laplace differential equation
ζu00(ζ) + (2−ζ)u0(ζ) − (1−ν)u(ζ) =0. (A.7) The general solution of the Laplace differential equation above is given in terms of the confluent hypergeometric function11F1, as
u(ζ) =c01F1(1−ν;2;ζ) +c1ζ−11F1(−ν;0;ζ). (A.8) Basic properties of the confluent hypergeometric function can be used to show that the expressionζ−11F1(−ν;0;ζ)diverges, and therefore it must be necessarilyc1=0. In the same manner, it can be shown that1F1(1−ν;2;ζ)is equal, up to a proportionality constant, to the associated Laguerre polynomial2 L1ν(ζ). Finally, the solution in terms of the original functionφ(z)is,
φν(z) =2c0 An inspection of eq. (A.9) reveals that the boundary conditionφ(0) =0is trivially satisfied; the conditionφ(z−→∞) =0on the other hand is satisfied, becauseL1ν(z)are polynomials increasing asO(zν), slower than the exponential decay of orderO(e−z). The normalization constantc0is determined by the conditionc20R∞
0 dz|φν(z)|2=1. The first three bound states can be seen in fig. A.1.
1We are referring to theMathematicafunction Hypergeometric1F1[a,b,z].
2Generalized, or associated Laguerre polynomials are called with the help of theMathematica function LaguerreL[a,b,z].
A.2. PERTURBATIVE SOLUTION IN THE PRESENCE OF HOLDING FIELD 139
The eigenenergiesEνcan be calculated from, Eν= −mΛ2
8h2 1
ν2≈−7.23kB
ν2 , ν=1, 2, . . . (A.10) It is very convenient to express the energiesEin terms of the temperature of the corresponding thermal energy resulting fromE=kBT. Then, the “tem-perature difference” between the first excited and ground state is3.6K. Since the temperature of TDES on the surface of liquid helium can be≈2K at most, all electrons will be found at the ground state and it is impossible to excite electrons thermally.
A.2. Perturbative solution in the presence of holding field
The presence of a holding electric fieldE⊥along thez-direction is enough to make Schrödinger equation intractable analytically. Approximate methods must be used and the most common is perturbation method. This method is applicable whenever the hamiltonian of the problem can be written in the form H^= ^H0+δV^, whereH^0possess an analytical solution andδV^can be considered a ‘small’ perturbation, in the sense thatδ1.
Assumingδ1, one can expand both the eigenenergiesE(s)ν and the eigen-functions|φ(s)ν iofH^ in a power series with respect toδ. Usually, the first order correction(s=1)is a very accurate approximation and it will be calculated by means of the formulas,
E(1)ν =E(0)ν +δhφν|V|φ^ νi, (A.11a)
|φ(1)ν i=|φ(0)ν i+δX
k6=ν
hφ(0)k |V|φ^ (0)ν i
E(0)ν −E(0)k |φ(0)k i. (A.11b) These formulas are valid only for nondegenerate eigenstates, because other-wise it might happen thatE(0)ν =E(0)k for two distinct eigenstates|φ(0)ν i,|φ(0)k i.
It is a common practice in experiments involving TDES on liquid helium to impose a static electric fieldE⊥ alongz-direction. The hamiltonian of an electron on the surface of liquid4He takes the form
H^ = −h2
2m∇2−Λ
z−eE⊥z, (A.12)
whereby it is immediately obvious thateE⊥zis a candidate perturbation term. It was shown thatH^ can be decomposed into a part involving only(x, y)and a part involving the heightz; the perturbation term affects only thez-coordinate, and any corrections will be therefore constrained to|φν(z)iand the corresponding eigenenergiesEν.
If the perturbation term is written as a function of the dimensionless param-eterζ(defined in eq. A.4), such thatV= −2eλE⊥ζ, then
δ= −2eλE⊥=⇒δ= −2eE⊥
νa ≈−8.20×10−11E⊥
ν [SI], (A.13) which is indeed much less than unity for every conceivable holding field magni-tudeE⊥, whose maximum value cannot be larger than104V/m.
Values ofhφ(0)k (ζ)|ζ|φ(0)ν (ζ)i
k=. . . 1 2 3 4
ν=1 12 −5√
2 √
3 0
ν=2 −5√
2 9 −7√
6/3 √ 2
An inspection of first order perturbation formulas eqs. (A.11) reveals already that the energy spectrum will be shifted to more negative values, but no splitting will occur because there is no degeneracy to be lifted in thez-direction. For obtaining quantitative results, expressions of the form
hφ(0)k |ζ|φ(0)ν i=ckcν
Z∞
0
dζ ζ3e−ζL1k(ζ)L1ν(ζ), (A.14) need to be calculated. The integration is easy to do and for reference purposes the values ofhφ(0)k |ζ|φ(0)ν ifor variousν, kare provided in tabular form. An extremely fortuitous result that should be kept in mind is that all other values fork > 4are equal to zero, and the summation in eq. (A.11b) is actually finite, consisting of only two terms for the ground state correction,ν=1.
APPENDIX B