Download Quasi-exactly solvable problems in Quantum Mechanics

Quasi-exactly solvable
problems in Quantum
Mechanics.
New recently derived
Hamiltonians.
Michael Kreshchuk,
PhD candidate at the University of Minnesota
• M. Shifman “Quasi-exactly solvable spectral problems”
• A.V. Turbiner “Quasi-Exactly-Solvable Problems and sl(2) Algebra”
• M. Kreshchuk “A quasi-exactly solvable model: two charges in a
magnetic field, subject to a non-Coulomb mutual interaction”
Solvable problems 1

2
Matrix QM: the procedure of solving the time-independent Schrödinger
equation
Hψ=Eψ
is equivalent to diagonalization of the infinite-dimensional matrix
Solvable problems 2

This can be done for a very small number of potentials.
A textbook example: harmonic oscillator
3
Solvable problems 3




Harmonic oscillator:
Morse potential:
𝜔𝜔2 2
𝑉𝑉 =
𝑥𝑥
2
𝑉𝑉 = 𝐴𝐴(𝑒𝑒 −2𝛼𝛼𝛼𝛼 − 𝑒𝑒 −𝛼𝛼𝛼𝛼 )
Pöschl-Teller potential:
𝑉𝑉 = −
𝑈𝑈0
cosh2 𝛼𝛼𝛼𝛼
Coulomb potential plus centrifugal barrier:
𝑉𝑉 = −
𝛼𝛼 𝑙𝑙 (𝑙𝑙 + 1)
+
2
2 𝑟𝑟 2
4
Quasi-exactly solvable Hamiltonian

5
There exist potentials which permit for partial diagonalization of
the Hamiltonian. They are called quasi-exactly solvable (QES):
6
Main idea
Assume that you have a finite-dimensional vector space V and a set of
operators Jk acting in V and mapping it into itself. So none of the
operators can take us out of the vector space:
∀𝑥𝑥 ∈ 𝑉𝑉, ∀𝐽𝐽𝑘𝑘 :
𝐽𝐽𝑘𝑘 𝑣𝑣 ∈ 𝑉𝑉

Example 1: (n x 1) vectors and (n x n) matrices.

Example 2: Ylm‘s (for a fixed l) and the angular momentum operators.
Then, any operator of the form
𝐴𝐴 = 𝐶𝐶0 + 𝐶𝐶𝑘𝑘 𝐽𝐽𝑘𝑘 + 𝐶𝐶𝑘𝑘𝑘𝑘 𝐽𝐽𝑘𝑘 𝐽𝐽𝑙𝑙 + ⋯
cannot drive us out of the space V either. This means that the associated
spectral problem
𝐴𝐴 𝑣𝑣 = 𝜆𝜆 𝑣𝑣
can be treated as a matrix problem and, therefore, can be reduced to
diagonalization of a finite-dimensional matrix.
SL(2,C) generators in the
differential form

Reminder:
7
[ J x , J y ] = iJ z
J + = J x + iJ y , J − = J x − iJ y , J 0 = J z

These generators can be represented by the differential operators
J + = ξ 2 ∂ ξ − 2 jξ , J − = ∂ ξ , J 0 = ξ∂ ξ − j
acting in a linear space spanned by the vectors {1, ξ, ξ 2,… ξ 2j}.

In contrast with the angular momentum operators, two of these
have an explicit j-dependence.
Main idea: once again

The process of solution of the equation
f ( J k ) p (ξ ) = λp (ξ )
can be reduced to diagonalization of a finite-dimensional
matrix, if we assume p(ξ) to be polynomials of the power 2j.

Indeed, the vector space spanned by {1, ξ, ξ 2,… ξ 2j} is
invariant under the action of Jk’s.

For powers greater than 2j , this is no longer true:
J +ξ 2 j = 0, but
J +ξ 2 j +1 = ξ 2 j + 2
8
9
Some adjustments (1-dimensional case)

The original Hamiltonian cannot be reduced to a nice form.
However, a simple transformation is of a great use here.

Reminder: the solution of the QHO is ~ 𝑝𝑝𝑛𝑛 𝑥𝑥 exp −𝑥𝑥 2 .


We start from the Schrödinger equation

,
and change variables as 𝜓𝜓 𝑥𝑥 → 𝑝𝑝𝑛𝑛 (𝑥𝑥); usually, the change is like
𝜓𝜓 = 𝑝𝑝𝑛𝑛 𝑥𝑥 exp(𝑞𝑞(𝑥𝑥))

𝐻𝐻 𝜓𝜓 = 𝐸𝐸 𝜓𝜓
or
𝜓𝜓 = 𝑝𝑝𝑛𝑛 𝑥𝑥 𝑥𝑥 𝑠𝑠 exp(𝑞𝑞(𝑥𝑥))
where 𝑞𝑞(𝑥𝑥) is a polynomial.
Thus, we switch to a new spectral problem
𝐻𝐻"𝐺𝐺𝐺 𝑝𝑝𝑛𝑛 𝑥𝑥 = 𝐸𝐸 ′ 𝑝𝑝𝑛𝑛 (𝑥𝑥) ,
where 𝐻𝐻"𝐺𝐺𝐺 сan be expressed through generators of SL(2,C).
13
Simplest QES potential 1

We introduce new variable ξ = x2. Then the equation
H "G " pn (ξ ) = E pn (ξ )

With Hamiltonian
H "G " = −2 J 0 J − − (2 j + 1) J − −ν J + + µ J 0
for the function pn(ξ) is equivalent to the equation
 1 d2

 ψ ( x) = E ψ ( x)
+
H ψ ( x) =  −
V
x
(
)
2
dx
2


for the function ψ(x), where pn(x) = ψ(x) exp(-ν x4/8+μ x2/4) .

With the potential
µ2
3 

V (x) =
x +
x +  − 2ν  j +  x 2
8
4
8 

8
ν2
6
µν
4
µ ,ν ∈ R
j = 1 / 2, 1, 3 / 2...
15
Simplest QES potential 2

Thus, the solutions of the Schrödinger equation have the form
µ 
 ν
ψ ( x) = p2 j ( x 2 ) exp - x 4 − x 2 
 8
4


where coefficients of the polynomial p2j are obtained via
diagonalization of the (2j+1)x(2j+1) matrix H”G”, and the energies
are its eigenvalues.

The way to construct the matrix is straightforward:
We act with the operator
H "G " = −2 J 0 J − − (2 j + 1) J − −ν J + + µ J 0
on the wave functions of the form
p 2 j (ξ ) = 1 + αξ + βξ 2 + ... + γξ 2 j
16
Facts 1

All known exactly-solvable potentials could, of course, be
represented in the form discussed above:
V in the original H

H”G”
Not a coincidence: none of them contains the raising operaor T+.
Facts 2

QES potentials, for the Schrödinger equation −𝜕𝜕𝑥𝑥2 + 𝑉𝑉 𝑥𝑥
17
𝜓𝜓 𝑥𝑥 = 𝐸𝐸 𝜓𝜓(𝑥𝑥)
The statement of the problem


18
Sometimes the Schrödinger equation looks different from
−𝜕𝜕𝑥𝑥2 + 𝑉𝑉 𝑥𝑥
𝜓𝜓 𝑥𝑥 = 𝐸𝐸 𝜓𝜓(𝑥𝑥)
The Coulomb problem is exactly-solvable. However, we can add
magnetic field. This case was considered by
A.V. Turbiner and M.A. Escobar-Ruiz

What are the other possibilities for the form of the mutual interaction in
the presence of the magnetic field?
19
The Coulomb case 1

The problem allows for quasi-exact solutions either when it is neutral
(e1 = - e2), or when e1/ m1= e2/ m2 (all the calculations are analogous).
,


The equation for the relative motion has form
After the separation and change of variables
one gets the following equation for the relative coordinate:
,
20
The Coulomb case 2


The latter can be written in terms of the SL(2,C) generators (n ≡ 2j) as follows:
𝑛𝑛
𝜖𝜖
−𝐽𝐽𝑛𝑛0 𝐽𝐽− + 𝐽𝐽𝑛𝑛+ − 1 + 2 𝑠𝑠 + 𝐽𝐽− 𝑝𝑝 𝜌𝜌 = −
𝑝𝑝 𝜌𝜌 = −𝜈𝜈 𝑝𝑝(𝜌𝜌)
𝜔𝜔𝑐𝑐 𝑚𝑚𝑟𝑟
2
𝑓𝑓 𝐽𝐽𝑛𝑛0 , 𝐽𝐽𝑛𝑛+ , 𝐽𝐽− 𝑝𝑝(𝜌𝜌) = −𝜈𝜈 𝑝𝑝(𝜌𝜌)
The price for quasi-exact solvability: an additional relation (‘constraint’ or
‘quantization condition’) for the parameters of the problem. The term
𝜖𝜖
𝜈𝜈 =
𝜔𝜔𝑐𝑐 𝑚𝑚𝑟𝑟
on the RHS has to take one of the n+1 values for the system to be QES.
Here
𝜖𝜖 =
2𝑚𝑚1 𝑚𝑚2 𝑒𝑒1 𝑒𝑒2
𝑚𝑚1 +𝑚𝑚2
,
𝜔𝜔𝑐𝑐 =
𝐵𝐵(𝑒𝑒1 +𝑒𝑒2 )
𝑚𝑚1 + 𝑚𝑚2
The Coulomb case 3

21
One way to treat the quantization condition
𝜖𝜖 = 𝜈𝜈 𝜔𝜔𝑐𝑐 𝑚𝑚𝑟𝑟
is to say that once we have fixed the masses and charges of
the particle, the magnetic field is ‘quantized’ in the sense that
there exists an infinite discrete set of values of the magnetic
field B for which the exact analytic solutions of the Schrödinger
equation exist.
1 4𝑚𝑚1 𝑚𝑚2 𝑒𝑒12 𝑒𝑒22
𝐵𝐵 = 2
𝜈𝜈 (𝑒𝑒1 + 𝑒𝑒2 )
(a set of n+1 ν’s for each n)
New potential 1

The ‘quantization condition’:
22
New potential 2

The ‘quantization condition’:
23
New potential 3

The ‘quantization condition’ (two of them):
24
Literature
28
1.
M Shifman 1998 Quasi-Exactly Solvable Spectral Problems
2.
A V Turbiner 1988 Quasi-Exactly-Solvable Problems and sl(2) Algebra
3.
A G Ushveridze 1994 Quasi-Exactly Solvable Models in Quantum
Mechanics
29
THANK YOU!