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Exact Solutions > Linear Partial Differential Equations >
Second-Order Parabolic Partial Differential Equations > Linear Schrodinger (Schrödinger) Equation
2
2
1.8. Schrodinger Equation ih̄ ∂w = – h̄ ∂ w2 + U (x)w
∂t
2m ∂x
1.8-1. Eigenvalue problem. Cauchy problem for the Schrodinger’s equation.
The Schrodinger’s (Schrödinger’s) equation is the basic equation of quantum mechanics; w is the
wave function, i2 = −1, h̄ is Planck’s constant, m is the mass of the particle, and U (x) is the potential
energy of the particle in the force field.
1◦ . In discrete spectrum problems, the particular solutions are sought in the form
µ
¶
iEn
w(x, t) = exp −
t ψn (x),
h̄
where the eigenfunctions ψn and the respective energies En have to be determined by solving the
eigenvalue problem
¤
d2 ψn 2m £
+ 2 En − U (x) ψn = 0,
dx2
h̄
(1)
Z ∞
ψn → 0 at x → ±∞,
|ψn |2 dx = 1.
−∞
The last relation is the normalizing condition for ψn .
2◦ . In the cases where the eigenfunctions ψn (x) form an orthonormal basis in L2 (R), the solution
of the Cauchy problem for Schrodinger’s equation with the initial condition
w = f (x) at
t=0
(2)
is given by
Z
w(x, t) =
∞
−∞
G(x, ξ, t)f (ξ) dξ,
G(x, ξ, t) =
∞
X
n=0
µ
¶
iEn
ψn (x)ψn (ξ) exp −
t .
h̄
Various potentials U (x) are considered below and particular solutions of the boundary value
problem (1) or the Cauchy problem for Schrodinger’s equation are presented.
1.8-2. Free particle: U (x) = 0.
The solution of the Cauchy problem for the Schrodinger’s equation with the initial condition (2) is
given by
·
½
¸
Z ∞
√
√
(x − ξ)2
1
h̄t
eπi/4 √|a| if a > 0,
exp −
w(x, t) = √
ia =
f (ξ) dξ,
τ=
,
4iτ
2m
e−πi/4 |a| if a < 0.
2 iπτ −∞
1.8-3. Linear potential (motion in a uniform external field): U (x) = ax.
Solution of the Cauchy problem for the Schrodinger’s equation with the initial condition (2):
·
¸
Z
¢ ∞
¡
(x + bτ 2 − ξ)2
1
h̄t
2am
exp −
w(x, t) = √
exp −ibτ x − 13 ib2 τ 3
f (ξ) dξ, τ =
, b= 2 .
4iτ
2m
2 iπτ
h̄
−∞
1
2
LINEAR SCHRODINGER (SCHRÖDINGER) EQUATION
1.8-4. Linear harmonic oscillator: U (x) =
Eigenvalues:
1
mω 2 x2 .
2
¡
¢
En = h̄ω n + 12 ,
n = 0, 1, . . .
Normalized eigenfunctions:
¡
¢
1
exp − 12 ξ 2 Hn (ξ),
ψn (x) = 1/4 √ n
π
2 n! x0
x
,
ξ=
x0
r
x0 =
h̄
,
mω
where Hn (ξ) are the Hermite polynomials. The functions ψn (x) form an orthonormal basis in
L2 (R).
1.8-5. Isotropic free particle: U (x) = a/x2 .
Here, the variable x ≥ 0 plays the role of the radial coordinate, and a > 0. The equation with
U (x) = a/x2 results from Schrodinger’s equation for a free particle with n space coordinates if one
passes to spherical (cylindrical) coordinates and separates the angular variables.
The solution of Schrodinger’s equation satisfying the initial condition (2) has the form
£
¤Z ∞
µ 2
¶ µ
¶
exp − 21 iπ(µ + 1) sign t
x + y2
xy
√
w(x, t) =
xy exp i
Jµ
f (y) dy,
2|τ |
4τ
2|τ |
0
r
2am 1
h̄t
, µ=
τ=
+ ≥ 1,
2m
4
h̄2
where Jµ (ξ) is the Bessel function.
1.8-6. Morse potential: U (x) = U0 (e–2x/a – 2e–x/a ).
Eigenvalues:
·
¸2
1
En = −U0 1 − (n + 21 ) ,
β
β=
√
a 2mU0
,
h̄
0 ≤ n < β − 2.
ξ = 2βe−x/a ,
s=
Eigenfunctions:
ψn (x) = ξ s e−ξ/2 Φ(−n, 2s + 1, ξ),
√
a −2mEn
,
h̄
where Φ(a, b, ξ) is the degenerate hypergeometric function.
In this case the number of eigenvalues (energy levels) En and eigenfunctions ψn is finite:
n = 0, 1, . . . , nmax .
References
Krein, S. G. (Editor), Functional Analysis [in Russian], Nauka, Moscow, 1972.
Landau, L. D. and Lifshits, E. M., Quantum Mechanics. Nonrelativistic Theory [in Russian], Nauka, Moscow, 1974.
Miller, W. (Jr.), Symmetry and Separation of Variables, Addison-Wesley, London, 1977.
Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics, Dover Publ., New York, 1990.
Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists , Chapman & Hall/CRC,
2002.
Linear Schrodinger (Schrödinger) Equation
c 2004 Andrei D. Polyanin
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