Probability current in the relativistic Hamiltonian quantum mechanics
Jakub Rembieli´nski
University ofL´od´z
Max Born Symposium, University of Wroclaw, 28–30 June, 2011 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010)
p ∂ H = c2p2 + m2c4 + V (x), H → i , x → x, p → −i ∇ ~∂t ~ The spinless Salpeter equation:
∂ p i φ(x, t) = [ m2c4 − 2c2∆ + V (x)]φ(x, t) ~∂t ~ L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102, 568 (1956)
Probability current 2/27 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010)
p ∂ H = c2p2 + m2c4 + V (x), H → i , x → x, p → −i ∇ ~∂t ~ The spinless Salpeter equation:
∂ p i φ(x, t) = [ m2c4 − 2c2∆ + V (x)]φ(x, t) ~∂t ~ L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102, 568 (1956)
Probability current 2/27 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010)
p ∂ H = c2p2 + m2c4 + V (x), H → i , x → x, p → −i ∇ ~∂t ~ The spinless Salpeter equation:
∂ p i φ(x, t) = [ m2c4 − 2c2∆ + V (x)]φ(x, t) ~∂t ~ L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102, 568 (1956)
Probability current 2/27 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of:
K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010)
p ∂ H = c2p2 + m2c4 + V (x), H → i , x → x, p → −i ∇ ~∂t ~ The spinless Salpeter equation:
∂ p i φ(x, t) = [ m2c4 − 2c2∆ + V (x)]φ(x, t) ~∂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 the integro-differential equation ∂φ(x, t) Z i = d 3y K(x − y)φ(y, t) + V (x)φ(x, t) ~ ∂t where the kernel is given by
2 3 mc 2m c K2( |x − y|) K(x − y) = − ~ (2π)2~ |x − y|2 The Salpeter equation presumes the Newton-Wigner localization scheme implying the standard quantization rule xˆ → x, pˆ → −i~∇. Consequently, the Hilbert space of solutions to the Salpeter is L2(R3, d 3x): Z hφ|ψi = d 3x φ∗(x)ψ(x)
Therefore, we should identify |φ(x, t)|2 with the probability density ρ(x, t) satisfying the normalization condition: Z d 3x ρ(x, t) = 1
Probability current 5/27 The Salpeter equation can be written in the form of the integro-differential equation ∂φ(x, t) Z i = d 3y K(x − y)φ(y, t) + V (x)φ(x, t) ~ ∂t where the kernel is given by
2 3 mc 2m c K2( |x − y|) K(x − y) = − ~ (2π)2~ |x − y|2 The Salpeter equation presumes the Newton-Wigner localization scheme implying the standard quantization rule xˆ → x, pˆ → −i~∇. Consequently, the Hilbert space of solutions to the Salpeter is L2(R3, d 3x): Z hφ|ψi = d 3x φ∗(x)ψ(x)
Therefore, we should identify |φ(x, t)|2 with the probability density ρ(x, t) satisfying the normalization condition: Z d 3x ρ(x, t) = 1
Probability current 5/27 The Salpeter equation can be written in the form of the integro-differential equation ∂φ(x, t) Z i = d 3y K(x − y)φ(y, t) + V (x)φ(x, t) ~ ∂t where the kernel is given by
2 3 mc 2m c K2( |x − y|) K(x − y) = − ~ (2π)2~ |x − y|2 The Salpeter equation presumes the Newton-Wigner localization scheme implying the standard quantization rule xˆ → x, pˆ → −i~∇. Consequently, the Hilbert space of solutions to the Salpeter is L2(R3, d 3x): Z hφ|ψi = d 3x φ∗(x)ψ(x)
Therefore, we should identify |φ(x, t)|2 with the probability density ρ(x, t) satisfying the normalization condition: Z d 3x 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description 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 light cone structure 2 3 3 I the space L (R , d x) of solutions to the Salpeter equation is invariant under the Lorentz group
I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation
I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation
I the Salpeter equation is widely used in the phenomenological description of the quark-antiquark-gluon system as a hadron model
Probability current 6/27 The Klein-Gordon equation
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
The Klein-Gordon equation was accepted because of its manifest Lorentz covariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can be negative)
I paradoxes such as “zitterbewegung” and Klein paradox connected with existence of negative energy solutions Nonrelativistic version of the Klein-Gordon theory would be
∂ψ pˆ2 ∂2ψ pˆ4 i = ψ → − 2 = ψ ~ ∂t 2m ~ ∂t2 4m2
Probability current 7/27 The Klein-Gordon equation
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
The Klein-Gordon equation was accepted because of its manifest Lorentz covariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can be negative)
I paradoxes such as “zitterbewegung” and Klein paradox connected with existence of negative energy solutions Nonrelativistic version of the Klein-Gordon theory would be
∂ψ pˆ2 ∂2ψ pˆ4 i = ψ → − 2 = ψ ~ ∂t 2m ~ ∂t2 4m2
Probability current 7/27 The Klein-Gordon equation
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
The Klein-Gordon equation was accepted because of its manifest Lorentz covariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can be negative)
I paradoxes such as “zitterbewegung” and Klein paradox connected with existence of negative energy solutions Nonrelativistic version of the Klein-Gordon theory would be
∂ψ pˆ2 ∂2ψ pˆ4 i = ψ → − 2 = ψ ~ ∂t 2m ~ ∂t2 4m2
Probability current 7/27 The Klein-Gordon equation
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
The Klein-Gordon equation was accepted because of its manifest Lorentz covariance. However its grave flaws are:
I problems with probablistic interpretation (probability density can be negative)
I paradoxes such as “zitterbewegung” and Klein paradox connected with existence of negative energy solutions Nonrelativistic version of the Klein-Gordon theory would be
∂ψ pˆ2 ∂2ψ pˆ4 i = ψ → − 2 = ψ ~ ∂t 2m ~ ∂t2 4m2
Probability current 7/27 Probability current
Probability current for the Klein-Gordon equation
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
In the case of the Klein-Gordon equation the probability density is
∗ i~ ∗ ∂ψ ∂ψ 2i 2 ρKG = ψ − ψ + V |ψ| 2mc2 ∂t ∂t ~ The corresponding probablity current is given by i j = − ~ (ψ∗∇ψ − ψ∇ψ∗) KG 2m 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
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
In the case of the Klein-Gordon equation the probability density is
∗ i~ ∗ ∂ψ ∂ψ 2i 2 ρKG = ψ − ψ + V |ψ| 2mc2 ∂t ∂t ~ The corresponding probablity current is given by i j = − ~ (ψ∗∇ψ − ψ∇ψ∗) KG 2m 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
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
In the case of the Klein-Gordon equation the probability density is
∗ i~ ∗ ∂ψ ∂ψ 2i 2 ρKG = ψ − ψ + V |ψ| 2mc2 ∂t ∂t ~ The corresponding probablity current is given by i j = − ~ (ψ∗∇ψ − ψ∇ψ∗) KG 2m 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
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
In the case of the Klein-Gordon equation the probability density is
∗ i~ ∗ ∂ψ ∂ψ 2i 2 ρKG = ψ − ψ + V |ψ| 2mc2 ∂t ∂t ~ The corresponding probablity current is given by i j = − ~ (ψ∗∇ψ − ψ∇ψ∗) KG 2m 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
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
In the case of the Klein-Gordon equation the probability density is
∗ i~ ∗ ∂ψ ∂ψ 2i 2 ρKG = ψ − ψ + V |ψ| 2mc2 ∂t ∂t ~ The corresponding probablity current is given by i j = − ~ (ψ∗∇ψ − ψ∇ψ∗) KG 2m 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
∂ 2 i − V (x) ψ(x, t) = (m2c4 − 2c2∆)ψ(x, t) ~∂t ~
In the case of the Klein-Gordon equation the probability density is
∗ i~ ∗ ∂ψ ∂ψ 2i 2 ρKG = ψ − ψ + V |ψ| 2mc2 ∂t ∂t ~ The corresponding probablity current is given by i j = − ~ (ψ∗∇ψ − ψ∇ψ∗) KG 2m Problems:
I ρKG can be negative
I difficulties with the limit m = 0
Probability current 8/27 Probability current for the Salpeter equation
∂ p i φ(x, t) = [ m2c4 − 2c2∆ + V (x)]φ(x, t) ~∂t ~ Probability density: ρ(x, t) = |φ(x, t)|2 Using the continuity equation ∂ρ + ∇·j = 0 ∂t we derived the following formula on the probability current:
Z (k−p)·x c 3 3 p + k i ∗ j(x, t) = d pd k e ~ φ˜ (p, t)φ˜(k, t) 3 6 p (2π) ~ pm2c2 + p2 + m2c2 + k2
Probability current 9/27 Probability current for the Salpeter equation
∂ p i φ(x, t) = [ m2c4 − 2c2∆ + V (x)]φ(x, t) ~∂t ~ Probability density: ρ(x, t) = |φ(x, t)|2 Using the continuity equation ∂ρ + ∇·j = 0 ∂t we derived the following formula on the probability current:
Z (k−p)·x c 3 3 p + k i ∗ j(x, t) = d pd k e ~ φ˜ (p, t)φ˜(k, t) 3 6 p (2π) ~ pm2c2 + p2 + m2c2 + k2
Probability current 9/27 Probability current for the Salpeter equation
∂ p i φ(x, t) = [ m2c4 − 2c2∆ + V (x)]φ(x, t) ~∂t ~ Probability density: ρ(x, t) = |φ(x, t)|2 Using the continuity equation ∂ρ + ∇·j = 0 ∂t we derived the following formula on the probability current:
Z (k−p)·x c 3 3 p + k i ∗ j(x, t) = d pd k e ~ φ˜ (p, t)φ˜(k, t) 3 6 p (2π) ~ pm2c2 + p2 + m2c2 + k2
Probability current 9/27 Properties:
I correct nonrelativistic limit i lim j = − ~ (φ∗∇φ − φ∇φ∗) c→∞ 2m
I good behaviour of the total current Z j(x, t) d 3x = hφ|vˆφi
where vˆ is the operator of the relativistic velocity
cpˆ vˆ = pˆ0 q √ 2 2 2 2 2 2 where pˆ = −i~∇, andp ˆ0 = E/c = m c + pˆ = m c − ~ ∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27 Properties:
I correct nonrelativistic limit i lim j = − ~ (φ∗∇φ − φ∇φ∗) c→∞ 2m
I good behaviour of the total current Z j(x, t) d 3x = hφ|vˆφi
where vˆ is the operator of the relativistic velocity
cpˆ vˆ = pˆ0 q √ 2 2 2 2 2 2 where pˆ = −i~∇, andp ˆ0 = E/c = m c + pˆ = m c − ~ ∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27 Properties:
I correct nonrelativistic limit i lim j = − ~ (φ∗∇φ − φ∇φ∗) c→∞ 2m
I good behaviour of the total current Z j(x, t) d 3x = hφ|vˆφi
where vˆ is the operator of the relativistic velocity
cpˆ vˆ = pˆ0 q √ 2 2 2 2 2 2 where pˆ = −i~∇, andp ˆ0 = E/c = m c + pˆ = m c − ~ ∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27 Properties:
I correct nonrelativistic limit i lim j = − ~ (φ∗∇φ − φ∇φ∗) c→∞ 2m
I good behaviour of the total current Z j(x, t) d 3x = hφ|vˆφi
where vˆ is the operator of the relativistic velocity
cpˆ vˆ = pˆ0 q √ 2 2 2 2 2 2 where pˆ = −i~∇, andp ˆ0 = E/c = m c + pˆ = m c − ~ ∆.
These formula is not valid in the case of the Klein-Gordon equation.
Probability current 10/27 hφ|vˆφi2 ≤ c2
I existence of the massless limit
Z (k−p)·x c 3 3 p + k i ∗ j(x, t) = d pd k e ~ φ˜ (p, t)φ˜(k, t)(m = 0) (2π)3~6 |p| + |k| We have a remarkable formula Z 3 3 ∗ ∗ j(x, t) = d yd z K(|y−x|, |x−z|)[φ (y, t)∇zφ(z, t)−φ(z, t)∇yφ (y, t)] where im2c3 1 1 hmc i K(|u|, |w|) = − K2 (|u| + |w|) (2π)3~2 |u||w| |u| + |w| ~ The current can be also written as
∞ 2n 2n−1 imc2 X (2n − 3)!! X j = − ~ (−1)k ∇k φ∗∇2n−k−1φ (2n)!! mc ~ n=1 k=0
Probability current 11/27 hφ|vˆφi2 ≤ c2
I existence of the massless limit
Z (k−p)·x c 3 3 p + k i ∗ j(x, t) = d pd k e ~ φ˜ (p, t)φ˜(k, t)(m = 0) (2π)3~6 |p| + |k| We have a remarkable formula Z 3 3 ∗ ∗ j(x, t) = d yd z K(|y−x|, |x−z|)[φ (y, t)∇zφ(z, t)−φ(z, t)∇yφ (y, t)] where im2c3 1 1 hmc i K(|u|, |w|) = − K2 (|u| + |w|) (2π)3~2 |u||w| |u| + |w| ~ The current can be also written as
∞ 2n 2n−1 imc2 X (2n − 3)!! X j = − ~ (−1)k ∇k φ∗∇2n−k−1φ (2n)!! mc ~ n=1 k=0
Probability current 11/27 hφ|vˆφi2 ≤ c2
I existence of the massless limit
Z (k−p)·x c 3 3 p + k i ∗ j(x, t) = d pd k e ~ φ˜ (p, t)φ˜(k, t)(m = 0) (2π)3~6 |p| + |k| We have a remarkable formula Z 3 3 ∗ ∗ j(x, t) = d yd z K(|y−x|, |x−z|)[φ (y, t)∇zφ(z, t)−φ(z, t)∇yφ (y, t)] where im2c3 1 1 hmc i K(|u|, |w|) = − K2 (|u| + |w|) (2π)3~2 |u||w| |u| + |w| ~ The current can be also written as
∞ 2n 2n−1 imc2 X (2n − 3)!! X j = − ~ (−1)k ∇k φ∗∇2n−k−1φ (2n)!! mc ~ n=1 k=0
Probability current 11/27 hφ|vˆφi2 ≤ c2
I existence of the massless limit
Z (k−p)·x c 3 3 p + k i ∗ j(x, t) = d pd k e ~ φ˜ (p, t)φ˜(k, t)(m = 0) (2π)3~6 |p| + |k| We have a remarkable formula Z 3 3 ∗ ∗ j(x, t) = d yd z K(|y−x|, |x−z|)[φ (y, t)∇zφ(z, t)−φ(z, t)∇yφ (y, t)] where im2c3 1 1 hmc i K(|u|, |w|) = − K2 (|u| + |w|) (2π)3~2 |u||w| |u| + |w| ~ The current can be also written as
∞ 2n 2n−1 imc2 X (2n − 3)!! X j = − ~ (−1)k ∇k φ∗∇2n−k−1φ (2n)!! mc ~ n=1 k=0
Probability current 11/27 Examples Free massless particle on a line r ∂φ(x, t) ∂2 i = − φ(x, t), ∂t ∂x 2 where we set c = 1.The solution is
r2a a + it φ(x, t) = π x 2 + (a + it)2
Hence, we get the probability density
2a a2 + t2 ρ(x, t) = |φ(x, t)|2 = π (x 2 − t2 + a2)2 + 4a2t2
and the probability current
a (x + t)2 + a2 ax x 2 − 3t2 + a2 j(x, t) = ln − 4πt2 (x − t)2 + a2 πt (x 2 − t2 + a2)2 + 4a2t2
Probability current 12/27 Examples Free massless particle on a line r ∂φ(x, t) ∂2 i = − φ(x, t), ∂t ∂x 2 where we set c = 1.The solution is
r2a a + it φ(x, t) = π x 2 + (a + it)2
Hence, we get the probability density
2a a2 + t2 ρ(x, t) = |φ(x, t)|2 = π (x 2 − t2 + a2)2 + 4a2t2
and the probability current
a (x + t)2 + a2 ax x 2 − 3t2 + a2 j(x, t) = ln − 4πt2 (x − t)2 + a2 πt (x 2 − t2 + a2)2 + 4a2t2
Probability current 12/27 Examples Free massless particle on a line r ∂φ(x, t) ∂2 i = − φ(x, t), ∂t ∂x 2 where we set c = 1.The solution is
r2a a + it φ(x, t) = π x 2 + (a + it)2
Hence, we get the probability density
2a a2 + t2 ρ(x, t) = |φ(x, t)|2 = π (x 2 − t2 + a2)2 + 4a2t2
and the probability current
a (x + t)2 + a2 ax x 2 − 3t2 + a2 j(x, t) = ln − 4πt2 (x − t)2 + a2 πt (x 2 − t2 + a2)2 + 4a2t2
Probability current 12/27 Examples Free massless particle on a line r ∂φ(x, t) ∂2 i = − φ(x, t), ∂t ∂x 2 where we set c = 1.The solution is
r2a a + it φ(x, t) = π x 2 + (a + it)2
Hence, we get the probability density
2a a2 + t2 ρ(x, t) = |φ(x, t)|2 = π (x 2 − t2 + a2)2 + 4a2t2
and the probability current
a (x + t)2 + a2 ax x 2 − 3t2 + a2 j(x, t) = ln − 4πt2 (x − t)2 + a2 πt (x 2 − t2 + a2)2 + 4a2t2
Probability current 12/27 10
5 t
0
0.6
0.4 ΡHx, tL
0.2
0.0 -20 -10 0 10 20 x
Figure: The time evolution of the probability density related to the solution of the Salpeter equation for a free massless particle in one dimension. The parameter a = 1.
Probability current 13/27 10
5 t
0 0.2
0.1
jHx, tL 0.0
-0.1
-0.2 -20 -10 0 10 20 x
Figure: The time development of the probability current for a free massless particle moving in a line. The parameter a = 1. Probability current 14/27 The solution referring to the particle moving to the right (left)
r a ±i φ (x, t) = ± π x ∓ t ± ia Therefore, the corresponding probability density and probability current are a 1 ρ (x, t) = |φ (x, t)|2 = ±j (x, t) = ± ± ± π (x ∓ t)2 + a2
Probability current 15/27 40
30
20 t
10
0
0.3
0.2 Ρ+Hx, tL
0.1
0.0 -20 0 20 40 x
Figure: The behavior of the probability density ρ+(x, t) referring to the case of the free massles particle moving to the right. The stable maximum of the probability density is going with the speed of light c=1.
Probability current 16/27 Free massive particle on a line r ∂φ(x, t) ∂2 i = m2 − φ(x, t) ∂t ∂x 2 We have found the solution r m a + it p 2 2 φ(x, t) = p K1[m x + (a + it) ] πK1(2ma) x 2 + (a + it)2
The corresponding probability current is
2 Z x 2 2 m x − (a + it) p 2 2 j(x, t) = dx Im 2 2 K2[m x + (a + it) ] πK1(2ma) 0 x + (a + it) i p 2 2 − K0[m x + (a + it) ] ) a − it p 2 2 × K1[m x + (a − it) ] px 2 + (a − it)2
Probability current 17/27 Free massive particle on a line r ∂φ(x, t) ∂2 i = m2 − φ(x, t) ∂t ∂x 2 We have found the solution r m a + it p 2 2 φ(x, t) = p K1[m x + (a + it) ] πK1(2ma) x 2 + (a + it)2
The corresponding probability current is
2 Z x 2 2 m x − (a + it) p 2 2 j(x, t) = dx Im 2 2 K2[m x + (a + it) ] πK1(2ma) 0 x + (a + it) i p 2 2 − K0[m x + (a + it) ] ) a − it p 2 2 × K1[m x + (a − it) ] px 2 + (a − it)2
Probability current 17/27 Free massive particle on a line r ∂φ(x, t) ∂2 i = m2 − φ(x, t) ∂t ∂x 2 We have found the solution r m a + it p 2 2 φ(x, t) = p K1[m x + (a + it) ] πK1(2ma) x 2 + (a + it)2
The corresponding probability current is
2 Z x 2 2 m x − (a + it) p 2 2 j(x, t) = dx Im 2 2 K2[m x + (a + it) ] πK1(2ma) 0 x + (a + it) i p 2 2 − K0[m x + (a + it) ] ) a − it p 2 2 × K1[m x + (a − it) ] px 2 + (a − it)2
Probability current 17/27 10
5 t
1.0 0
ΡHx, tL 0.5
0.0 -10 -5 0 5 10 x
Figure: The time development of the probability density ρ(x, t) = |φ(x, t)|2 corresponding to the case of the free massive particle. The mass m = 0.5 and a = 1.
Probability current 18/27 30
20 t 10
0 0.2
0.1
jHx, tL 0.0
-0.1
-0.2 -20 0 20 x
Figure: The plot of the probability current for a free massive particle moving in a line versus time, where m = 0.5 and a = 1.
Probability current 19/27 Massless particle in a linear potential r ∂φ(x, t) ∂2 i = − φ(x, t) + xφ(x, t) ∂t ∂x 2 where we set c = 1 and ~ = 1. We have found the solution of the form 1 ( " # 4 (−λt+ix)2 1 λ − λ t2 1 −λt + ix φ(x, t) = e 2 √ e 2(λ+i) erfc 2 π λ + i p2(λ + i) " #) 1 (λt−ix)2 λt − ix + √ e 2(λ−i) erfc λ − i p2(λ − i)
z 2 √2 R −t where erfc(z) = 1 − π 0 e dt is the complementary error function.
Probability current 20/27 Massless particle in a linear potential r ∂φ(x, t) ∂2 i = − φ(x, t) + xφ(x, t) ∂t ∂x 2 where we set c = 1 and ~ = 1. We have found the solution of the form 1 ( " # 4 (−λt+ix)2 1 λ − λ t2 1 −λt + ix φ(x, t) = e 2 √ e 2(λ+i) erfc 2 π λ + i p2(λ + i) " #) 1 (λt−ix)2 λt − ix + √ e 2(λ−i) erfc λ − i p2(λ − i)
z 2 √2 R −t where erfc(z) = 1 − π 0 e dt is the complementary error function.
Probability current 20/27 20
10
0 t
-10
-20 0.6
0.4 ΡHx, tL 0.2
0.0 -30 -20 -10 0 10 x
Figure: The plot of the probability density ρ(x, t) = |φ(x, t)|2, referring to the massless particle in a linear potential. The parameter λ = 1. The classical Λ-shaped dynamics of the maxima of the probability density is easily observed. Probability current 21/27 ` YxHtL] t -2 -1 1 2
-0.5
-1.0
-1.5
-2.0
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 10
0 t
-10
0.5
jHx, tL 0.0
-0.5 -20 -10 0 x
Figure: The plot of the probability current versus time.
Probability current 23/27 Plane wave solutions ∂φ(x, t) p i = m2c4 − 2c2∆ φ(x, t) ~ ∂t ~ possesses plane wave solutions
− i (Et−k·x) φ(x, t) = Ce ~ p 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
c2k v = E
Probability current 24/27 Plane wave solutions ∂φ(x, t) p i = m2c4 − 2c2∆ φ(x, t) ~ ∂t ~ possesses plane wave solutions
− i (Et−k·x) φ(x, t) = Ce ~ p 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
c2k v = E
Probability current 24/27 Plane wave solutions ∂φ(x, t) p i = m2c4 − 2c2∆ φ(x, t) ~ ∂t ~ possesses plane wave solutions
− i (Et−k·x) φ(x, t) = Ce ~ p 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
c2k v = E
Probability current 24/27 Massless particle in three dimensions
∂φ(x, t) √ i = −∆ φ(x, t) ∂t We have obtained the following solution
3 (2a) 2 a + it φ(x, t) = π [r 2 + (a + it)2]2 where r = |x|. Therefore the probability density is
(2a)3 a2 + t2 ρ(x, t) = |φ(x, t)|2 = π2 [(r 2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
a3 32r 2t4 + (r 2 + t2 + a2)[3(r 2 − t2 + a2)2 + 12a2t2 − 8r 2t2] j(x, t) = − 2π2r 2t3 [(r 2 − t2 + a2)2 + 4a2t2]2 3a3 (r + t)2 + a2 + ln x 8π2r 3t4 (r − t)2 + a2
Probability current 25/27 Massless particle in three dimensions
∂φ(x, t) √ i = −∆ φ(x, t) ∂t We have obtained the following solution
3 (2a) 2 a + it φ(x, t) = π [r 2 + (a + it)2]2 where r = |x|. Therefore the probability density is
(2a)3 a2 + t2 ρ(x, t) = |φ(x, t)|2 = π2 [(r 2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
a3 32r 2t4 + (r 2 + t2 + a2)[3(r 2 − t2 + a2)2 + 12a2t2 − 8r 2t2] j(x, t) = − 2π2r 2t3 [(r 2 − t2 + a2)2 + 4a2t2]2 3a3 (r + t)2 + a2 + ln x 8π2r 3t4 (r − t)2 + a2
Probability current 25/27 Massless particle in three dimensions
∂φ(x, t) √ i = −∆ φ(x, t) ∂t We have obtained the following solution
3 (2a) 2 a + it φ(x, t) = π [r 2 + (a + it)2]2 where r = |x|. Therefore the probability density is
(2a)3 a2 + t2 ρ(x, t) = |φ(x, t)|2 = π2 [(r 2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
a3 32r 2t4 + (r 2 + t2 + a2)[3(r 2 − t2 + a2)2 + 12a2t2 − 8r 2t2] j(x, t) = − 2π2r 2t3 [(r 2 − t2 + a2)2 + 4a2t2]2 3a3 (r + t)2 + a2 + ln x 8π2r 3t4 (r − t)2 + a2
Probability current 25/27 Massless particle in three dimensions
∂φ(x, t) √ i = −∆ φ(x, t) ∂t We have obtained the following solution
3 (2a) 2 a + it φ(x, t) = π [r 2 + (a + it)2]2 where r = |x|. Therefore the probability density is
(2a)3 a2 + t2 ρ(x, t) = |φ(x, t)|2 = π2 [(r 2 − t2 + a2)2 + 4a2t2]2
The probability current is given by
a3 32r 2t4 + (r 2 + t2 + a2)[3(r 2 − t2 + a2)2 + 12a2t2 − 8r 2t2] j(x, t) = − 2π2r 2t3 [(r 2 − t2 + a2)2 + 4a2t2]2 3a3 (r + t)2 + a2 + ln x 8π2r 3t4 (r − t)2 + a2
Probability current 25/27 1.0
0.5 t
1.0 0.0
ΡHr, tL 0.5
0.0 0.0 0.5 1.0 1.5 2.0 r
Figure: The time evolution of the probability density, where a = 1, showing the spreading of the wavefunction.
Probability current 26/27 15
10 t
5
0.004
jHr, tL¤ 0.002
0.000 0 5 10 15 20 r
Figure: The time development of the norm of the probability current, where a = 1.
Probability current 27/27