Probability current in the relativistic Hamiltonianquantum mechanics
Jakub Rembielinski
University of Lodz
Max Born Symposium, University of Wroc law, 28–30 June, 2011
Formulation of relativistic quantum mechanics: Salpeterequation
K. Kowalski and J. Rembielinski, Salpeter equation and probabilitycurrent in the relativistic Hamiltonian quantum mechanics (accepted forpublication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembielinski, The relativistic massless harmonicoscillator, Phys. Rev. A 81, 012118 (2010)
H =√
c2p2 + m2c4 + V (x), H → i~∂
∂t, x→ x, p→ −i~∇
The spinless Salpeter equation:
i~∂
∂tφ(x, t) = [
√m2c4 − ~2c2∆ + V (x)]φ(x, t)
L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102,568 (1956)
Probability current 2/27
Formulation of relativistic quantum mechanics: Salpeterequation
K. Kowalski and J. Rembielinski, Salpeter equation and probabilitycurrent in the relativistic Hamiltonian quantum mechanics (accepted forpublication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembielinski, The relativistic massless harmonicoscillator, Phys. Rev. A 81, 012118 (2010)
H =√
c2p2 + m2c4 + V (x), H → i~∂
∂t, x→ x, p→ −i~∇
The spinless Salpeter equation:
i~∂
∂tφ(x, t) = [
√m2c4 − ~2c2∆ + V (x)]φ(x, t)
L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102,568 (1956)
Probability current 2/27
Formulation of relativistic quantum mechanics: Salpeterequation
K. Kowalski and J. Rembielinski, Salpeter equation and probabilitycurrent in the relativistic Hamiltonian quantum mechanics (accepted forpublication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembielinski, The relativistic massless harmonicoscillator, Phys. Rev. A 81, 012118 (2010)
H =√
c2p2 + m2c4 + V (x), H → i~∂
∂t, x→ x, p→ −i~∇
The spinless Salpeter equation:
i~∂
∂tφ(x, t) = [
√m2c4 − ~2c2∆ + V (x)]φ(x, t)
L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102,568 (1956)
Probability current 2/27
Formulation of relativistic quantum mechanics: Salpeterequation
K. Kowalski and J. Rembielinski, Salpeter equation and probabilitycurrent in the relativistic Hamiltonian quantum mechanics (accepted forpublication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembielinski, The relativistic massless harmonicoscillator, Phys. Rev. A 81, 012118 (2010)
H =√
c2p2 + m2c4 + V (x), H → i~∂
∂t, x→ x, p→ −i~∇
The spinless Salpeter equation:
i~∂
∂tφ(x, t) = [
√m2c4 − ~2c2∆ + V (x)]φ(x, t)
L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102,568 (1956)
Probability current 2/27
Leslie L. Foldy (1919–2001)
Probability current 3/27
Edwin E. Salpeter (1924–2008)Probability current 4/27
The Salpeter equation can be written in the form of theintegro-differential equation
i~∂φ(x, t)
∂t=
∫d3yK (x− y)φ(y, t) + V (x)φ(x, t)
where the kernel is given by
K (x− y) = − 2m2c3
(2π)2~K2(mc
~ |x− y|)|x− y|2
The Salpeter equation presumes the Newton-Wigner localization schemeimplying the standard quantization rule x→ x, p→ −i~∇.Consequently, the Hilbert space of solutions to the Salpeter isL2(R3, d3x):
〈φ|ψ〉 =
∫d3xφ∗(x)ψ(x)
Therefore, we should identify |φ(x, t)|2 with the probability densityρ(x, t) satisfying the normalization condition:∫
d3x ρ(x, t) = 1
Probability current 5/27
The Salpeter equation can be written in the form of theintegro-differential equation
i~∂φ(x, t)
∂t=
∫d3yK (x− y)φ(y, t) + V (x)φ(x, t)
where the kernel is given by
K (x− y) = − 2m2c3
(2π)2~K2(mc
~ |x− y|)|x− y|2
The Salpeter equation presumes the Newton-Wigner localization schemeimplying the standard quantization rule x→ x, p→ −i~∇.Consequently, the Hilbert space of solutions to the Salpeter isL2(R3, d3x):
〈φ|ψ〉 =
∫d3xφ∗(x)ψ(x)
Therefore, we should identify |φ(x, t)|2 with the probability densityρ(x, t) satisfying the normalization condition:∫
d3x ρ(x, t) = 1
Probability current 5/27
The Salpeter equation can be written in the form of theintegro-differential equation
i~∂φ(x, t)
∂t=
∫d3yK (x− y)φ(y, t) + V (x)φ(x, t)
where the kernel is given by
K (x− y) = − 2m2c3
(2π)2~K2(mc
~ |x− y|)|x− y|2
The Salpeter equation presumes the Newton-Wigner localization schemeimplying the standard quantization rule x→ x, p→ −i~∇.Consequently, the Hilbert space of solutions to the Salpeter isL2(R3, d3x):
〈φ|ψ〉 =
∫d3xφ∗(x)ψ(x)
Therefore, we should identify |φ(x, t)|2 with the probability densityρ(x, t) satisfying the normalization condition:∫
d3x ρ(x, t) = 1
Probability current 5/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Salpeter equation was discarded because of:
I its nonlocality
I the lack of manifest Lorentz covariance
However
I the nonlocality of the Salpeter equation does not disturb the lightcone structure
I the space L2(R3, d3x) of solutions to the Salpeter equation isinvariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies onlyand we have no problems with paradoxes occuring in the case of theKlein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation withthe experimental spectrum of mesonic atoms is as good as for theKlein-Gordon equation
I the Salpeter equation is widely used in the phenomenologicaldescription of the quark-antiquark-gluon system as a hadron model
Probability current 6/27
The Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
The Klein-Gordon equation was accepted because of its manifest Lorentzcovariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can benegative)
I paradoxes such as “zitterbewegung” and Klein paradox connectedwith existence of negative energy solutions
Nonrelativistic version of the Klein-Gordon theory would be
i~∂ψ
∂t=
p2
2mψ → −~2 ∂
2ψ
∂t2=
p4
4m2ψ
Probability current 7/27
The Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
The Klein-Gordon equation was accepted because of its manifest Lorentzcovariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can benegative)
I paradoxes such as “zitterbewegung” and Klein paradox connectedwith existence of negative energy solutions
Nonrelativistic version of the Klein-Gordon theory would be
i~∂ψ
∂t=
p2
2mψ → −~2 ∂
2ψ
∂t2=
p4
4m2ψ
Probability current 7/27
The Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
The Klein-Gordon equation was accepted because of its manifest Lorentzcovariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can benegative)
I paradoxes such as “zitterbewegung” and Klein paradox connectedwith existence of negative energy solutions
Nonrelativistic version of the Klein-Gordon theory would be
i~∂ψ
∂t=
p2
2mψ → −~2 ∂
2ψ
∂t2=
p4
4m2ψ
Probability current 7/27
The Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
The Klein-Gordon equation was accepted because of its manifest Lorentzcovariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can benegative)
I paradoxes such as “zitterbewegung” and Klein paradox connectedwith existence of negative energy solutions
Nonrelativistic version of the Klein-Gordon theory would be
i~∂ψ
∂t=
p2
2mψ → −~2 ∂
2ψ
∂t2=
p4
4m2ψ
Probability current 7/27
Probability current
Probability current for the Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
In the case of the Klein-Gordon equation the probability density is
ρKG =i~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t+
2i
~V |ψ|2
)The corresponding probablity current is given by
jKG = − i~2m
(ψ∗∇ψ − ψ∇ψ∗)
Problems:
I ρKG can be negative
I difficulties with the limit m = 0
Probability current 8/27
Probability current
Probability current for the Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
In the case of the Klein-Gordon equation the probability density is
ρKG =i~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t+
2i
~V |ψ|2
)The corresponding probablity current is given by
jKG = − i~2m
(ψ∗∇ψ − ψ∇ψ∗)
Problems:
I ρKG can be negative
I difficulties with the limit m = 0
Probability current 8/27
Probability current
Probability current for the Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
In the case of the Klein-Gordon equation the probability density is
ρKG =i~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t+
2i
~V |ψ|2
)The corresponding probablity current is given by
jKG = − i~2m
(ψ∗∇ψ − ψ∇ψ∗)
Problems:
I ρKG can be negative
I difficulties with the limit m = 0
Probability current 8/27
Probability current
Probability current for the Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
In the case of the Klein-Gordon equation the probability density is
ρKG =i~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t+
2i
~V |ψ|2
)The corresponding probablity current is given by
jKG = − i~2m
(ψ∗∇ψ − ψ∇ψ∗)
Problems:
I ρKG can be negative
I difficulties with the limit m = 0
Probability current 8/27
Probability current
Probability current for the Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
In the case of the Klein-Gordon equation the probability density is
ρKG =i~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t+
2i
~V |ψ|2
)The corresponding probablity current is given by
jKG = − i~2m
(ψ∗∇ψ − ψ∇ψ∗)
Problems:
I ρKG can be negative
I difficulties with the limit m = 0
Probability current 8/27
Probability current
Probability current for the Klein-Gordon equation[i~∂
∂t− V (x)
]2
ψ(x, t) = (m2c4 − ~2c2∆)ψ(x, t)
In the case of the Klein-Gordon equation the probability density is
ρKG =i~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t+
2i
~V |ψ|2
)The corresponding probablity current is given by
jKG = − i~2m
(ψ∗∇ψ − ψ∇ψ∗)
Problems:
I ρKG can be negative
I difficulties with the limit m = 0
Probability current 8/27
Probability current for the Salpeter equation
i~∂
∂tφ(x, t) = [
√m2c4 − ~2c2∆ + V (x)]φ(x, t)
Probability density:ρ(x, t) = |φ(x, t)|2
Using the continuity equation
∂ρ
∂t+∇·j = 0
we derived the following formula on the probability current:
j(x, t) =c
(2π)3~6
∫d3pd3k
p + k√m2c2 + p2 +
√m2c2 + k2
ei(k−p)·x
~ φ∗(p, t)φ(k, t)
Probability current 9/27
Probability current for the Salpeter equation
i~∂
∂tφ(x, t) = [
√m2c4 − ~2c2∆ + V (x)]φ(x, t)
Probability density:ρ(x, t) = |φ(x, t)|2
Using the continuity equation
∂ρ
∂t+∇·j = 0
we derived the following formula on the probability current:
j(x, t) =c
(2π)3~6
∫d3pd3k
p + k√m2c2 + p2 +
√m2c2 + k2
ei(k−p)·x
~ φ∗(p, t)φ(k, t)
Probability current 9/27
Probability current for the Salpeter equation
i~∂
∂tφ(x, t) = [
√m2c4 − ~2c2∆ + V (x)]φ(x, t)
Probability density:ρ(x, t) = |φ(x, t)|2
Using the continuity equation
∂ρ
∂t+∇·j = 0
we derived the following formula on the probability current:
j(x, t) =c
(2π)3~6
∫d3pd3k
p + k√m2c2 + p2 +
√m2c2 + k2
ei(k−p)·x
~ φ∗(p, t)φ(k, t)
Probability current 9/27
Properties:
I correct nonrelativistic limit
limc→∞
j = − i~2m
(φ∗∇φ− φ∇φ∗)
I good behaviour of the total current∫j(x, t) d3x = 〈φ|vφ〉
where v is the operator of the relativistic velocity
v =cp
p0
where p = −i~∇, and p0 = E/c =
√m2c2 + p2 =
√m2c2 − ~2∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27
Properties:
I correct nonrelativistic limit
limc→∞
j = − i~2m
(φ∗∇φ− φ∇φ∗)
I good behaviour of the total current∫j(x, t) d3x = 〈φ|vφ〉
where v is the operator of the relativistic velocity
v =cp
p0
where p = −i~∇, and p0 = E/c =
√m2c2 + p2 =
√m2c2 − ~2∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27
Properties:
I correct nonrelativistic limit
limc→∞
j = − i~2m
(φ∗∇φ− φ∇φ∗)
I good behaviour of the total current∫j(x, t) d3x = 〈φ|vφ〉
where v is the operator of the relativistic velocity
v =cp
p0
where p = −i~∇, and p0 = E/c =
√m2c2 + p2 =
√m2c2 − ~2∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27
Properties:
I correct nonrelativistic limit
limc→∞
j = − i~2m
(φ∗∇φ− φ∇φ∗)
I good behaviour of the total current∫j(x, t) d3x = 〈φ|vφ〉
where v is the operator of the relativistic velocity
v =cp
p0
where p = −i~∇, and p0 = E/c =
√m2c2 + p2 =
√m2c2 − ~2∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27
〈φ|vφ〉2 ≤ c2
I existence of the massless limit
j(x, t) =c
(2π)3~6
∫d3pd3k
p + k
|p|+ |k|ei
(k−p)·x~ φ∗(p, t)φ(k, t) (m = 0)
We have a remarkable formula
j(x, t) =
∫d3yd3zK (|y−x|, |x−z|)[φ∗(y, t)∇zφ(z, t)−φ(z, t)∇yφ
∗(y, t)]
where
K (|u|, |w|) = − im2c3
(2π)3~2
1
|u||w|1
|u|+ |w|K2
[mc
~(|u|+ |w|)
]The current can be also written as
j = − imc2
~
∞∑n=1
(2n − 3)!!
(2n)!!
(~mc
)2n 2n−1∑k=0
(−1)k∇kφ∗∇2n−k−1φ
Probability current 11/27
〈φ|vφ〉2 ≤ c2
I existence of the massless limit
j(x, t) =c
(2π)3~6
∫d3pd3k
p + k
|p|+ |k|ei
(k−p)·x~ φ∗(p, t)φ(k, t) (m = 0)
We have a remarkable formula
j(x, t) =
∫d3yd3zK (|y−x|, |x−z|)[φ∗(y, t)∇zφ(z, t)−φ(z, t)∇yφ
∗(y, t)]
where
K (|u|, |w|) = − im2c3
(2π)3~2
1
|u||w|1
|u|+ |w|K2
[mc
~(|u|+ |w|)
]The current can be also written as
j = − imc2
~
∞∑n=1
(2n − 3)!!
(2n)!!
(~mc
)2n 2n−1∑k=0
(−1)k∇kφ∗∇2n−k−1φ
Probability current 11/27
〈φ|vφ〉2 ≤ c2
I existence of the massless limit
j(x, t) =c
(2π)3~6
∫d3pd3k
p + k
|p|+ |k|ei
(k−p)·x~ φ∗(p, t)φ(k, t) (m = 0)
We have a remarkable formula
j(x, t) =
∫d3yd3zK (|y−x|, |x−z|)[φ∗(y, t)∇zφ(z, t)−φ(z, t)∇yφ
∗(y, t)]
where
K (|u|, |w|) = − im2c3
(2π)3~2
1
|u||w|1
|u|+ |w|K2
[mc
~(|u|+ |w|)
]The current can be also written as
j = − imc2
~
∞∑n=1
(2n − 3)!!
(2n)!!
(~mc
)2n 2n−1∑k=0
(−1)k∇kφ∗∇2n−k−1φ
Probability current 11/27
〈φ|vφ〉2 ≤ c2
I existence of the massless limit
j(x, t) =c
(2π)3~6
∫d3pd3k
p + k
|p|+ |k|ei
(k−p)·x~ φ∗(p, t)φ(k, t) (m = 0)
We have a remarkable formula
j(x, t) =
∫d3yd3zK (|y−x|, |x−z|)[φ∗(y, t)∇zφ(z, t)−φ(z, t)∇yφ
∗(y, t)]
where
K (|u|, |w|) = − im2c3
(2π)3~2
1
|u||w|1
|u|+ |w|K2
[mc
~(|u|+ |w|)
]The current can be also written as
j = − imc2
~
∞∑n=1
(2n − 3)!!
(2n)!!
(~mc
)2n 2n−1∑k=0
(−1)k∇kφ∗∇2n−k−1φ
Probability current 11/27
ExamplesFree massless particle on a line
i∂φ(x , t)
∂t=
√− ∂2
∂x2φ(x , t),
where we set c = 1.The solution is
φ(x , t) =
√2a
π
a + it
x2 + (a + it)2
Hence, we get the probability density
ρ(x , t) = |φ(x , t)|2 =2a
π
a2 + t2
(x2 − t2 + a2)2 + 4a2t2
and the probability current
j(x , t) =a
4πt2ln
(x + t)2 + a2
(x − t)2 + a2− ax
πt
x2 − 3t2 + a2
(x2 − t2 + a2)2 + 4a2t2
Probability current 12/27
ExamplesFree massless particle on a line
i∂φ(x , t)
∂t=
√− ∂2
∂x2φ(x , t),
where we set c = 1.The solution is
φ(x , t) =
√2a
π
a + it
x2 + (a + it)2
Hence, we get the probability density
ρ(x , t) = |φ(x , t)|2 =2a
π
a2 + t2
(x2 − t2 + a2)2 + 4a2t2
and the probability current
j(x , t) =a
4πt2ln
(x + t)2 + a2
(x − t)2 + a2− ax
πt
x2 − 3t2 + a2
(x2 − t2 + a2)2 + 4a2t2
Probability current 12/27
ExamplesFree massless particle on a line
i∂φ(x , t)
∂t=
√− ∂2
∂x2φ(x , t),
where we set c = 1.The solution is
φ(x , t) =
√2a
π
a + it
x2 + (a + it)2
Hence, we get the probability density
ρ(x , t) = |φ(x , t)|2 =2a
π
a2 + t2
(x2 − t2 + a2)2 + 4a2t2
and the probability current
j(x , t) =a
4πt2ln
(x + t)2 + a2
(x − t)2 + a2− ax
πt
x2 − 3t2 + a2
(x2 − t2 + a2)2 + 4a2t2
Probability current 12/27
ExamplesFree massless particle on a line
i∂φ(x , t)
∂t=
√− ∂2
∂x2φ(x , t),
where we set c = 1.The solution is
φ(x , t) =
√2a
π
a + it
x2 + (a + it)2
Hence, we get the probability density
ρ(x , t) = |φ(x , t)|2 =2a
π
a2 + t2
(x2 − t2 + a2)2 + 4a2t2
and the probability current
j(x , t) =a
4πt2ln
(x + t)2 + a2
(x − t)2 + a2− ax
πt
x2 − 3t2 + a2
(x2 − t2 + a2)2 + 4a2t2
Probability current 12/27
-20 -10 0 10 20
x
0
5
10
t
0.0
0.2
0.4
0.6
ΡHx, tL
Figure: The time evolution of the probability density related to the solution ofthe Salpeter equation for a free massless particle in one dimension. Theparameter a = 1.
Probability current 13/27
-20 -10 0 10 20
x
0
5
10
t
-0.2
-0.1
0.0
0.1
0.2
jHx, tL
Figure: The time development of the probability current for a free masslessparticle moving in a line. The parameter a = 1.
Probability current 14/27
The solution referring to the particle moving to the right (left)
φ±(x , t) =
√a
π
±ix ∓ t ± ia
Therefore, the corresponding probability density and probability currentare
ρ±(x , t) = |φ±(x , t)|2 = ±j±(x , t) =a
π
1
(x ∓ t)2 + a2
Probability current 15/27
-20 0 20 40
x
0
10
20
30
40
t
0.0
0.1
0.2
0.3
Ρ+Hx, tL
Figure: The behavior of the probability density ρ+(x , t) referring to the case ofthe free massles particle moving to the right. The stable maximum of theprobability density is going with the speed of light c=1.
Probability current 16/27
Free massive particle on a line
i∂φ(x , t)
∂t=
√m2 − ∂2
∂x2φ(x , t)
We have found the solution
φ(x , t) =
√m
πK1(2ma)
a + it√x2 + (a + it)2
K1[m√x2 + (a + it)2]
The corresponding probability current is
j(x , t) =m2
πK1(2ma)
∫ x
0
dx Im
{[x2 − (a + it)2
x2 + (a + it)2K2[m
√x2 + (a + it)2]
− K0[m√x2 + (a + it)2]
]× a− it√
x2 + (a− it)2K1[m
√x2 + (a− it)2]
}
Probability current 17/27
Free massive particle on a line
i∂φ(x , t)
∂t=
√m2 − ∂2
∂x2φ(x , t)
We have found the solution
φ(x , t) =
√m
πK1(2ma)
a + it√x2 + (a + it)2
K1[m√x2 + (a + it)2]
The corresponding probability current is
j(x , t) =m2
πK1(2ma)
∫ x
0
dx Im
{[x2 − (a + it)2
x2 + (a + it)2K2[m
√x2 + (a + it)2]
− K0[m√x2 + (a + it)2]
]× a− it√
x2 + (a− it)2K1[m
√x2 + (a− it)2]
}
Probability current 17/27
Free massive particle on a line
i∂φ(x , t)
∂t=
√m2 − ∂2
∂x2φ(x , t)
We have found the solution
φ(x , t) =
√m
πK1(2ma)
a + it√x2 + (a + it)2
K1[m√x2 + (a + it)2]
The corresponding probability current is
j(x , t) =m2
πK1(2ma)
∫ x
0
dx Im
{[x2 − (a + it)2
x2 + (a + it)2K2[m
√x2 + (a + it)2]
− K0[m√x2 + (a + it)2]
]× a− it√
x2 + (a− it)2K1[m
√x2 + (a− it)2]
}
Probability current 17/27
-10 -5 0 5 10
x
0
5
10
t
0.0
0.5
1.0
ΡHx, tL
Figure: The time development of the probability density ρ(x , t) = |φ(x , t)|2corresponding to the case of the free massive particle. The mass m = 0.5 anda = 1.
Probability current 18/27
-20 0 20
x
0
10
20
30
t
-0.2
-0.1
0.0
0.1
0.2
jHx, tL
Figure: The plot of the probability current for a free massive particle moving ina line versus time, where m = 0.5 and a = 1.
Probability current 19/27
Massless particle in a linear potential
i∂φ(x , t)
∂t=
√− ∂2
∂x2φ(x , t) + xφ(x , t)
where we set c = 1 and ~ = 1. We have found the solution of the form
φ(x , t) =1
2
(λ
π
) 14
e−λ2 t
2
{1√λ+ i
e(−λt+ix)2
2(λ+i) erfc
[−λt + ix√
2(λ+ i)
]
+1√λ− i
e(λt−ix)2
2(λ−i) erfc
[λt − ix√2(λ− i)
]}
where erfc(z) = 1− 2√π
∫ z
0e−t
2
dt is the complementary error function.
Probability current 20/27
Massless particle in a linear potential
i∂φ(x , t)
∂t=
√− ∂2
∂x2φ(x , t) + xφ(x , t)
where we set c = 1 and ~ = 1. We have found the solution of the form
φ(x , t) =1
2
(λ
π
) 14
e−λ2 t
2
{1√λ+ i
e(−λt+ix)2
2(λ+i) erfc
[−λt + ix√
2(λ+ i)
]
+1√λ− i
e(λt−ix)2
2(λ−i) erfc
[λt − ix√2(λ− i)
]}
where erfc(z) = 1− 2√π
∫ z
0e−t
2
dt is the complementary error function.
Probability current 20/27
-30 -20 -10 0 10
x
-20
-10
0
10
20
t
0.0
0.2
0.4
0.6
ΡHx, tL
Figure: The plot of the probability density ρ(x , t) = |φ(x , t)|2, referring to themassless particle in a linear potential. The parameter λ = 1. The classicalΛ-shaped dynamics of the maxima of the probability density is easily observed.
Probability current 21/27
-2 -1 1 2t
-2.0
-1.5
-1.0
-0.5
Yx`
HtL]
Figure: The plot of the expectation value of the position operator versus time(solid line). The dotted line refers to the classical trajectory
Probability current 22/27
-20 -10 0
x
-10
0
10
t
-0.5
0.0
0.5
jHx, tL
Figure: The plot of the probability current versus time.
Probability current 23/27
Plane wave solutions
i~∂φ(x, t)
∂t=√m2c4 − ~2c2∆φ(x, t)
possesses plane wave solutions
φ(x, t) = Ce−i~ (Et−k·x)
where E =√m2c4 + k2c2 and C is a normalization constant. We have
j = ρv
where ρ = |φ|2 = |C |2, and v is the relativistic three-velocity given by
v =c2k
E
Probability current 24/27
Plane wave solutions
i~∂φ(x, t)
∂t=√m2c4 − ~2c2∆φ(x, t)
possesses plane wave solutions
φ(x, t) = Ce−i~ (Et−k·x)
where E =√m2c4 + k2c2 and C is a normalization constant. We have
j = ρv
where ρ = |φ|2 = |C |2, and v is the relativistic three-velocity given by
v =c2k
E
Probability current 24/27
Plane wave solutions
i~∂φ(x, t)
∂t=√m2c4 − ~2c2∆φ(x, t)
possesses plane wave solutions
φ(x, t) = Ce−i~ (Et−k·x)
where E =√m2c4 + k2c2 and C is a normalization constant. We have
j = ρv
where ρ = |φ|2 = |C |2, and v is the relativistic three-velocity given by
v =c2k
E
Probability current 24/27
Massless particle in three dimensions
i∂φ(x, t)
∂t=√−∆φ(x, t)
We have obtained the following solution
φ(x, t) =(2a)
32
π
a + it
[r2 + (a + it)2]2
where r = |x|. Therefore the probability density is
ρ(x, t) = |φ(x, t)|2 =(2a)3
π2
a2 + t2
[(r2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
j(x, t) =
{− a3
2π2r2t3
32r2t4 + (r2 + t2 + a2)[3(r2 − t2 + a2)2 + 12a2t2 − 8r2t2]
[(r2 − t2 + a2)2 + 4a2t2]2
+3a3
8π2r3t4ln
(r + t)2 + a2
(r − t)2 + a2
}x
Probability current 25/27
Massless particle in three dimensions
i∂φ(x, t)
∂t=√−∆φ(x, t)
We have obtained the following solution
φ(x, t) =(2a)
32
π
a + it
[r2 + (a + it)2]2
where r = |x|. Therefore the probability density is
ρ(x, t) = |φ(x, t)|2 =(2a)3
π2
a2 + t2
[(r2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
j(x, t) =
{− a3
2π2r2t3
32r2t4 + (r2 + t2 + a2)[3(r2 − t2 + a2)2 + 12a2t2 − 8r2t2]
[(r2 − t2 + a2)2 + 4a2t2]2
+3a3
8π2r3t4ln
(r + t)2 + a2
(r − t)2 + a2
}x
Probability current 25/27
Massless particle in three dimensions
i∂φ(x, t)
∂t=√−∆φ(x, t)
We have obtained the following solution
φ(x, t) =(2a)
32
π
a + it
[r2 + (a + it)2]2
where r = |x|. Therefore the probability density is
ρ(x, t) = |φ(x, t)|2 =(2a)3
π2
a2 + t2
[(r2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
j(x, t) =
{− a3
2π2r2t3
32r2t4 + (r2 + t2 + a2)[3(r2 − t2 + a2)2 + 12a2t2 − 8r2t2]
[(r2 − t2 + a2)2 + 4a2t2]2
+3a3
8π2r3t4ln
(r + t)2 + a2
(r − t)2 + a2
}x
Probability current 25/27
Massless particle in three dimensions
i∂φ(x, t)
∂t=√−∆φ(x, t)
We have obtained the following solution
φ(x, t) =(2a)
32
π
a + it
[r2 + (a + it)2]2
where r = |x|. Therefore the probability density is
ρ(x, t) = |φ(x, t)|2 =(2a)3
π2
a2 + t2
[(r2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
j(x, t) =
{− a3
2π2r2t3
32r2t4 + (r2 + t2 + a2)[3(r2 − t2 + a2)2 + 12a2t2 − 8r2t2]
[(r2 − t2 + a2)2 + 4a2t2]2
+3a3
8π2r3t4ln
(r + t)2 + a2
(r − t)2 + a2
}x
Probability current 25/27
0.0 0.5 1.0 1.5 2.0
r
0.0
0.5
1.0
t
0.0
0.5
1.0
ΡHr, tL
Figure: The time evolution of the probability density, where a = 1, showing thespreading of the wavefunction.
Probability current 26/27
0 5 10 15 20
r
5
10
15
t
0.000
0.002
0.004
jHr, tL¤
Figure: The time development of the norm of the probability current, wherea = 1.
Probability current 27/27