Eigenvalues for a Pure Quartic Oscillator
Author
S. M. Blinder
Title
Eigenvalues for a Pure Quartic Oscillator
Description
Computation on quartic oscillator using an operator formalism.
Category
Academic Articles & Supplements
Keywords
quartic oscillator, ladder operators, quantum mechanics
URL
http://www.notebookarchive.org/2019-03-9q5khpu/
DOI
https://notebookarchive.org/2019-03-9q5khpu
Date Added
2019-03-21
Date Last Modified
2019-03-21
File Size
130.32 kilobytes
Supplements
Rights
CC BY 4.0
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Eigenvalues for a Pure Quartic Oscillator
Eigenvalues for a Pure Quartic Oscillator
S. M. Blinder
Wolfram Research Inc.
Abstract
The eigenvalues of a pure quartic oscillator are computed, applying a canonical operator formulation, generalized from the harmonic oscillator. Solving a 10×10 secular equation produces eigenvalues in agreement, to at least 4 significant figures, with accurate computations given in the literature.
21 March 2019
Introduction
Introduction
The oscillator with a quartic anharmonicity, with Hamiltonian
H=+k+λ
2
p
2μ
1
2
2
x
1
4
4
x
(
1
)has been extensively treated in the literature [2,3]. This note will consider the pure quartic oscillator, in which the quadratic term is missing: . For simplicity, we take, . The Schr\bödinger equation we consider thus reduces to
k=0
μ=1
ℏ=1
-
λψ(x)=Eψ(x).
1
2
ψ''(x)+ 1 4 |
4
x
(
2
)The coordinate substitution transforms the equation into
xX
-1/6
λ
-
Ψ(X)=EΨ(X).
1
2
Ψ''(X)+ 1 4 |
4
X
-1/3
λ
(
3
)Thus only the case need be considered, with the energy scaling as for . No analytic solution of this problem has been found, but a number of accurate numerical computations have been carried out [4,5]. It is useful for general orientation to compare the WKB computation of the eigenvalues. The requisite equation is
λ=1
1/3
λ
λ≠1
2
E-
dx=2λ
4
4
x
2
1
4
(4E/λ)
∫
-
1
4
(4E/λ)
E-
dx=λ
4
4
x
16
3
-1/4
λ
3/4
E
1
2
(
4
)Where is a complete elliptic integral of the first kind. This gives the energies
K(·)
E
n
3/4
3
2
π
2/3
2
8/3
Γ
1
4
4/3
n+
1
2
1/3
λ
4/3
n+
1
2
1/3
λ
(
5
)The first 10 WKB energies, for , are tabulated below:= 0.344127, = 1.48895, = 2.94224, = 4.60804, = 6.44231,= 8.41864, =10.519, = 12.7303, = 15:0424, = 17.4471.
λ=1
E
0
E
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
E
9
Operator Formalism
Operator Formalism
We can obtain an accurate approximation to the eigenvalues of the quartic oscillator by an operator method, previously applied to the inversion of ammonia [6]. Since the Hamiltonian contains only even powers of and , a representation based on the ladder operators and suggests itself, a generalization of the canonical operator formulation for the harmonic oscillator. Accordingly, we define
p
x
a
†
a
a=x+ip,=x-ip
ω
2
1
2ω
†
a
ω
2
1
2ω
(
6
)The parameter is introduced, with its value to be determined such as to optimize the results. The actions of the ladder operators on a basis ket are given by
ω
a=
n
〈n-1〉,†
a
n+1
〈n+1〉.(
7
)so that
xn〉=
1
2ω
n
n-1+n+1
n+1(
8
)and
pn〉=
ω
2
n
n-1-n+1
n+1.(
9
)By successive application of these operators, it follows that
2
x
1
2ω
n(n-1)
n-2+(2n+1)n+(n+1)(n-2)
n+2(
10
) and
2
p
ω
2
n(n-1)
n-2-(2n+1)n+(n+1)(n-2)
n+2(
11
)Note, incidentally, that
2
p
2
1
2
2
ω
2
x
1
2
(
12
)which agrees with the result for an harmonic oscillator. Finally, we require
4
x
1
4
2
ω
n(n-1)(n-2)(n-3)
n-4+2n(n-1)
(2n-1)n-2+(62
n
(n+1)(n+2)
(2n+3)n+2+(n+1)(n+2)(n+3)(n+4)
n+4.(
13
)The nonzero matrix elements of the Hamiltonian are given by
H
n,n
ω
4
1
16
2
ω
2
n
H
n+2,n
H
n,n+2
ω
4
(n+1)(n+2)
+1
8
2
ω
(n+1)(n+2)
(2n+3),H
n+4,n
H
n,n+4
1
16
2
ω
(n+1)(n+2)(n+3)(n+4)
.(
14
)Results
Results
Following is the Mathematica program which produced the numerical results:
In[]:=
ω=2.16;
In[]:=
H[n_,m_]:=Whichmn,(2n+1)+(6+6n+3),mn+2,-
ω
4
1
16
2
ω
2
n
ω
4
(n+1)(n+2)
+1
8
2
ω
(n+1)(n+2)
(2n+3),mn+4,1
16
2
ω
(n+1)(n+2)(n+3)(n+4)
,m≠n||m≠n+2||m≠n+4,0In[]:=
h[n_,m_]:=H[Min[n,m],Max[n,m]]
In[]:=
mx=Array[h,{10,10},0];
In[]:=
eval=Reverse[Eigenvalues[mx]]
Out[]=
{0.420805,1.5079,2.95886,4.62127,6.46063,8.43686,10.6016,12.876,15.3116,17.7303}
Tabulating the lowest 10 eigenvalues:
In[]:=
E
0
E
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
E
9
These computations agree with the published results for the quartic oscillator to at least 4 significant figures. To summarize, we show a plot of the potential energy of a quartic oscillator, , on which is superposed the computed energies ,,…,, as horizontal red lines. For comparison the corresponding WKB energies are also shown as gray lines.
V(x)=
1
4
2
x
E
0
E
1
E
9
In[]:=
References
[1] A preliminary version of this computation: S. M. Blinder “Eigenvalues for a Pure Quartic Oscillator” http://demonstrations.wolfram.com/EigenvaluesForAPureQuarticOscillator/ Wolfram Demonstrations Project Published: March 15, 2019
[2] F. T. Hioe and E. W. Montroll, “Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity,” Journal of Mathematical Physics, 16, 1945 (1975).
[3] S. Mandal, “Quantum oscillator of quartic anharmonicity,” Journal of Physics A: Mathematical and General, 31, L501 (1998).
[4] S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava and V. S. Varma, “Eigenvalues of anharmonic oscillators,” Journal of Mathematical Physics, 14, 1190 (1973).
[5] P. M. Mathews, M. Seetharaman, S. Rraghavan and V. T. A. Bhargava, “A Simple Accurate Formula for the Energy Levels of Oscillators with a Quartic Potential,” Physics Letters, 83A(3), 118 (1981).
[6] S. M. Blinder, Ammonia Inversion Energy Levels using Operator Algebra, arXiv:1809.08178 (2018).
[2] F. T. Hioe and E. W. Montroll, “Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity,” Journal of Mathematical Physics, 16, 1945 (1975).
[3] S. Mandal, “Quantum oscillator of quartic anharmonicity,” Journal of Physics A: Mathematical and General, 31, L501 (1998).
[4] S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava and V. S. Varma, “Eigenvalues of
λ
2m
x
[5] P. M. Mathews, M. Seetharaman, S. Rraghavan and V. T. A. Bhargava, “A Simple Accurate Formula for the Energy Levels of Oscillators with a Quartic Potential,” Physics Letters, 83A(3), 118 (1981).
[6] S. M. Blinder, Ammonia Inversion Energy Levels using Operator Algebra, arXiv:1809.08178 (2018).
Cite this as: S. M. Blinder, "Eigenvalues for a Pure Quartic Oscillator" from the Notebook Archive (2018), https://notebookarchive.org/2019-03-9q5khpu
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