An extension of Shr ödinger’s quantization on the space (x, p), where the Hamiltonian approach is needed, is made on the space (x, v) where the Hamiltonian approach is not needed at all. The purpose of this paper is to give a possible extension of the actual formulation of the Quantum Mechanics, and this is achieved through a function K (x, v, t) which takes the place of the Hamiltonian on the Shr ödinger’s equation and has units of energy. This approach allows us to include the quantization of classical velocity depending problems (dissipative) and position depending mass variation problems. Some examples are given.
Despite the enormous success of the Hamilton formulation of the Quantum Mechanics [
In this work, it is proposed to make the quantization in terms of the variables ( x , v ) , where v = x ˙ , instead of the variables ( x , p ) , where p is the generalized linear momentum deduced from a Lagrangian of the system, p j = ∂ L / ∂ x ˙ j j = 1 , 2 , 3 , in fact, this was done firstly by Heisenberg at his beginning of the matrix theory quantization. In this way, one can get rid of the Hamiltonian formulation, and the goal is to obtain a function K ( x , v , t ) (having the energy as units) that can take the place of the Hamiltonian H ( x , p , t ) in the Schrödinger’s equation.
In this section, the analysis of several classical examples and cases will be made to obtain a function K ( x , v , t ) that can be used for quantization of the classical system in terms of its variables x and v .
Consider a conservative system which describes the motion in the space of a particle of mass position depending, m ( x ) , under a position depending force F ( x ) . Its Newton’s equation of motion is
d d t ( m 0 v ) = F ( x ) . (1)
This type of systems are invariant under Galileo’s transformations, as it is well known, and the so called energy is a constant of motion of the system,
K ( x , v ) = 1 2 m 0 v 2 − ∫ F ( x ) ⋅ d x , (2)
where the first term represents the kinetic energy, and the second one is the potential energy.
Consider a conservative system which describes the motion in the space of a particle of mass position depending, m ( x ) , under a position depending force F ( x ) . Its Newton’s equation of motion is
d d t ( m ( x ) v ) = F ( x ) . (3)
One must point out that this type of systems are not invariant under Galileo’s transformation [
1 2 ( m ( x ) v ) 2 = ∫ m ( x ) F ( x ) ⋅ v d t + C , (4)
where C is the integration constant. Knowing that one has the relation v d t = d x , and dividing the above expression by a characteristic mass m 0 , the following constant of motion is gotten
K ( x , v ) = ( m ( x ) v ) 2 2 m 0 − 1 m 0 ∫ m ( x ) F ( x ) ⋅ d x . (5)
The mass m 0 is that one that whenever m ( x ) = constant , this constant is m 0 . For example, for the 1-D harmonic oscillator ( F ( x ) = − m 0 ω 0 x 2 ) with mass variation m ( x ) = m 0 + m 1 x , and at first order in gradient m 1 , one would have the 1-D constant of motion
K ( x , v ) = 1 2 m 0 v 2 + 1 2 m 0 ω 0 2 x 2 + m 1 [ x v 2 + m 0 ω 0 2 3 x 3 ] , (6)
which will be a good approximation on the region | m 1 x | ≪ m 0 .
Consider the 1-D motion of a particle of constant mass m under the Hook’s force − k x in a dissipative medium which produces a velocity depending force of the form − α v , where k is the spring constant, and α is the dissipative constant. The associated dynamical system is given by the pair of equations
d x d t = v , d v d t = − ω 2 x − α m v , (7)
where ω = k / m represents the natural spring angular frequency. It has been shown [
K ( x , v ) = m 2 ( v 2 + 2 ω α x v + ω 2 x 2 ) e − 2 ω α G α ( v / x , ω ) , (8)
where ω α and G α are defined as
ω α = α / m , and G α = ( 1 2 Ω α ln ( ω α + v / x − Ω α ω α + v / x − Ω α ) , if ω < ω α 1 ω α + v / x , if ω = ω α 1 ω 2 − ω α 2 arctan ( ω α + v / x ω 2 − ω α 2 ) , if ω > ω α , (9)
where Ω α = ω α 2 − ω 2 , and corresponding to strong, critical, and weak dissipation cases. Of course, when dissipation is zero ( α = 0 ) one gets the usual energy of the harmonic oscillator.
Consider the motion of a particle with position mass depending m ( x ) under gravitational force, − m ( x ) g , in a dissipative medium where the force depende quadratically on its velocity, − α v 2 (with v < 0 ). The equation of motion is given by
d ( m ( x ) v ) d t = − m ( x ) g − α v 2 , (10)
where α is a non negative constant. This equation can be written as
m ( x ) d v d t = − m ( x ) g − ( α + m x ) v 2 , (11)
where m x is the variation of the mass with respect the position, m x = d m / d x . Integrating this equation, it is not difficult to see that one gets the following function
K ( x , v ) = ( m ( x ) v ) 2 2 m 0 e 2 α ∫ x d σ m ( σ ) + g m 0 ∫ x d σ m 2 ( σ ) e 2 α ∫ σ d s m ( s ) (12)
which is a constant of motion ( d K / d t = 0 ) of the system with energy units. The mass m 0 has been chosen such that if m ( x ) = constant , this constant has the value m 0 .
The motion of a charged particle of charge q and mass m under an electric and magnetic fields E and B is governed by Lorentz’s equation of motion [
d ( m v ) d t = q E + q c v × B , (13)
where c is the speed of light. This equation can be written in terms of the scalar potential Φ and vector potential A , where B = ∇ × A and E = − ∇ Φ − ∂ A / ∂ ( c t ) , as
d d t ( m v + q c A ) = − ∇ ( q Φ − q c A ⋅ v ) . (14)
Defining the new quantity of motion m V as
m V = m v + q A / c , (15)
and knowing that the function K associates to the equation d ( m V ) / d t = − ∇ ϕ is just K = m V 2 / 2 + ϕ , where V 2 = V x 2 + V y 2 + V z 2 , one can define the function K for this system as
K ( x , v , t ) = m 2 ( v + q m c A ) 2 + q Φ − q c A ⋅ v . (16)
For most of the cases, one has that A ⋅ v = 0 . So, the function K is given by
K ( x , v , t ) = m 2 ( v + q m c A ) 2 + q Φ . (17)
This case is given as a good example for completeness of the concept of a constant of motion. The equation of motion of a relativistic particle of constant mass m on a conservative force F ( x ) , with potential function V ( x ) = − ∫ F ( x ) ⋅ d x , is given by
d ( γ m v ) d t = F ( x ) , (18)
where the function γ is γ = ( 1 − | v | 2 / c 2 ) − 1 / 2 . It is well known [
K ( x , v ) = ( γ − 1 ) m c 2 + V ( x ) , (19)
where the function V ( x ) represents the potential energy of the system, V ( x ) = − ∫ F ( x ) ⋅ d x .
The main idea is to assign a linear operator to the energy function K ( x , v , t ) in order to have an equation that can be identify with Schrödinger’s equation. Firstly one needs to point out that in the case of Hamilton approach, if the generalized linear momentum, p j = ∂ L / ∂ v j with L ( x , v , t ) being a Lagrangian of the system, is related with the velocity of the form p j = m v j (m is the constant mass), the function K is exactly a Hamiltonian of the system, H ( x , p , t ) . Therefore, the quantization (Schrödinger’s equation) done with the function K can represent an extension of the theory of Quantum Mechanics. To do this, operators associated to the variables “ x ” and “ v ” are introduced, and as one could expect, the operators associated to these variables are postulated as
x → x ^ and v → v ^ = − i ℏ m ( x ) ∇ , (20)
where ℏ is as usual the Planck’s constant divided by 2π. In addition, one has the following commutation relation between the components of these operators
[ x k , v ^ l ] = − i ℏ m ( x ) δ k l I , (21)
where “I ” is the identity operator and the component v ^ l is
v ^ l = − i ℏ m ( x ) ∂ ∂ x l . (22)
One needs to point out that the operator (20) would be an Hermitian operator if and only if the mass does not depend on the position. The square, or self composition of this operator, has the following expression
v ^ 2 = − ℏ 2 m ( x ) ( − 1 m 2 ( x ) ∇ m ⋅ ∇ + 1 m ( x ) ∇ 2 ) . (23)
Then, using the above definitions, one associates and operator to the energy function K ( x , v , t ) as K ^ ( x , v ^ , t ) = i ℏ ∂ / ∂ t and defines the Schrödinger-like equation in the space ( x , v ) as
i ℏ ∂ Ψ ∂ t = K ^ ( x , v ^ , t ) Ψ , (24)
where Ψ = Ψ ( x , t ) is the wave function. If one has position depending mass problem, the operator K ^ may be not Hermitian ( K ^ † ≠ K ^ ) and there would not be conservation of the probability since the system is not invariant under Galileo’s transformation,
i ℏ d d t ∫ ℜ 3 | Ψ ( x , t ) | 2 d 3 x = 〈 Ψ , K ^ Ψ 〉 − 〈 K ^ † Ψ , Ψ 〉 , (25)
where the inner product has the usual definition, 〈 ϕ , ψ 〉 = ∫ ϕ † ( x ) ψ ( x ) d x .
However, for the case of position depending mass problems, one can construct a Schrödinger equation with Hermitian operator in the following way
i ℏ ∂ Ψ ∂ t = [ K ^ ( x , v ^ , t ) + K ^ † ( x , v ^ , t ) ] Ψ . (26)
In addition, although there might be some doubts where the function K ( x , v , t ) can not be a constant of motion [
Using (20) and (24) in (5), and assigning to the function f ( x ) 2 v 2 (for any arbitrary functionf) the operator
f 2 v 2 ^ = 1 3 ( v ^ 2 f 2 + f 2 v ^ 2 + v ^ f 2 v ^ ) , (27)
one can get the operator for the function m 2 v 2 as
m 2 v 2 ^ = − ℏ 2 3 m [ ( − 1 m 2 ∇ m ⋅ ∇ + 1 m ∇ 2 ) m 2 + 2 m ∇ 2 ] . (28)
Therefore, it follows that the Scrödinger-like equation is
i ℏ ∂ Ψ ∂ t = − ℏ 2 6 m 0 m ( x ) [ ( − 1 m ( x ) ∇ m ( x ) ⋅ ∇ + 1 m ( x ) ∇ 2 ) m 2 ( x ) + 2 m ( x ) ∇ 2 ] Ψ − [ 1 m 0 ∫ m ( x ) F ( x ) ⋅ d x ] Ψ . (29)
If one makes the approximation to an Hermitian operator v ^ = − ( i ℏ / m 0 ) ∇ , the operatior m 2 v 2 ^ would be given instead by
m 2 ( x ) v 2 ^ = − ℏ 2 3 m 0 2 [ 3 m ( x ) 2 ∇ 2 + 5 m ( x ) ∇ m ( x ) ⋅ ∇ + 2 ∇ m ( x ) ⋅ ∇ m ( x ) + 2 m ( x ) ( ∇ 2 m ( x ) ) ] , (30)
and the wave equation would be given by
i ℏ ∂ Ψ ∂ t = − ℏ 2 6 m 0 3 [ 3 m ( x ) 2 ∇ 2 + 5 m ( x ) ∇ m ( x ) ⋅ ∇ + 2 ∇ m ( x ) ⋅ ∇ m ( x ) + 2 m ( x ) ( ∇ 2 m ( x ) ) ] Ψ − [ 1 m 0 ∫ m ( x ) F ( x ) ⋅ d x ] Ψ . (31)
From the expression (17) and since the mass is constant, the quantization of these systems can be carried out with the following Shrödinger-like equation
i ℏ ∂ Ψ ∂ t = { m 2 ( − i ℏ m ∇ + q m c A ) 2 + q Φ } Ψ , (32)
where Ψ = Ψ ( x , t ) is an scalar function. Here again there is conservation of probability, and if the system has dipole electric and magnetic moments P and m , the interaction with the electric and magnetic fields can be added in the usual way [
[ σ k , σ j ] = i 2 ε k j l σ l , σ j 2 = I , ( σ → ⋅ a ) ( σ → ⋅ b ) = a ⋅ b I + i ( a × b ) ⋅ σ , S = ℏ 2 σ → , (33)
where I is the 2 × 2 identity matrix, σ → = ( σ x , σ y , σ z ) , a , b ∈ ℜ 3 are arbitrary vectors, and Einstein’s convention was used. Thus, the Pauli’s equation in the quantum space ( x , v ) for a charged particle of spin one-half can be written as
i ℏ ∂ Ψ ∂ t = { m 2 [ σ → ⋅ ( − i ℏ m ∇ + q m c A ) ] 2 + q Φ } Ψ , (34)
where Ψ is an spinor (a two components vector of scalar complex functions)
Ψ ( x , t ) = ( ψ 1 ( x , t ) ψ 2 ( x , t ) ) . (35)
From the expression (12), one notices that it will appear the product of f ( x ) v 2 where the this function is given by
f ( x ) = m 2 ( x ) e 2 α ∫ d σ / m ( σ ) . (36)
So, using the same expression (27) for 1-D, the Shcrödinger-like equation can be given as
i ℏ ∂ Ψ ∂ t = { − ℏ 2 6 m 0 [ ( − 1 m 2 ( x ) ∂ ∂ x m ( x ) ∂ ∂ x + 1 m ( x ) ∂ 2 ∂ x 2 ) m 2 ( x ) λ ( x ) + λ ( x ) ( − ∂ ∂ x m ( x ) ∂ ∂ x + m ( x ) ∂ 2 ∂ x 2 ) + ∂ ∂ x m ( x ) λ ( x ) ∂ ∂ x ] + g m 0 ∫ d σ m 2 ( σ ) λ ( σ ) } Ψ , (37)
where the function λ ( x ) has been defined as
λ ( x ) = e 2 α ∫ d s / m ( s ) . (38)
If the mass of the system is constant ( m ( x ) = m 0 ), one would have that v ^ = − ( i ℏ / m 0 ) ∂ / ∂ x is an Hermitian operator, the function λ would be λ ( x ) = e 2 α x / m 0 , and the wave equation would be
i ℏ ∂ Ψ ∂ t = { − ℏ 2 λ ( x ) 3 m 0 [ 3 ∂ 2 ∂ x 2 + ( 6 α m 0 ) ∂ ∂ x + ( 2 α m 0 ) 2 ] + g m 0 2 2 α ( λ ( x ) − 1 ) } Ψ . (39)
For completeness, for the system characterized by the expression (19), let us make first some algebraic manipulation. Let us write this expression in the form
K − V ( x ) + m c 2 = γ m c 2 ,
let us take the square of this expression and pass the velocity dependence to the left hand side. So, one gets
( 1 − v 2 c 2 ) ( K − V ( x ) + m c 2 ) 2 = m 2 c 4 . (40)
In this way, using the identification of the operators for different variables and the function K, it follows that
( 1 + ℏ 2 m 2 c 2 ∇ 2 ) ( − i ℏ ∂ ∂ t − V ( x ) + m c 2 ) 2 Ψ = m 2 c 4 Ψ . (41)
In this section, two simple solutions of the above approach are presented for illustration.
Using the approximated constant of motion (6), one can write this expression of the form
K ( x , v ) = K 0 ( x , v ) + W ( x , v ) , (42)
where K 0 represents the usual harmonic oscillator with constant mass m 0 , and W represents the term of the variation of mass at first approximation in Taylor expansion,
K 0 ( x , v ) = 1 2 m 0 v 2 + 1 2 m 0 ω 0 2 x 2 , W ( x , v ) = m 1 [ x v 2 + m 0 ω 0 2 3 x 3 ] . (43)
Of course, for m 1 = 0 , one has that p = m v , and K 0 represents the Hamiltonian H ( x , p ) = p 2 / 2 m 0 + m 0 ω 0 2 x 2 / 2 , and one knows that the solution of the Schrödinger’s equation would be Ψ 0 ( x , t ) = ∑ n = 0 c n exp ( − i E n t / ℏ ) Φ n ( x ) , where the set { E n = ℏ ω 0 ( n + 1 / 2 ) , Φ n ( x ) } is the solution of the eigenvalue problem H ^ Φ = E Φ [
i ℏ ∂ Ψ 0 ∂ t = K ^ ( x , v ^ ) Ψ 0 . (44)
For m 1 ≠ 0 , one considers that W is a perturbation of the system and uses perturbative theory to find the modification of the energy levels of the system. Since W contains odd monomials order, there is not contribution a first order perturbation, 〈 n | W ^ | n 〉 with 〈 x | n 〉 = Φ n ( x ) . It is not difficult to calculate that up to second order in perturbation theory, the eigenvalues are of the form
E n ≈ ℏ ω 0 ( n + 1 / 2 ) + m 1 2 ℏ 2 2 m 0 3 ( 39 ( 2 n 2 + 2 n ) 18 + 7 6 ) . (45)
Just to have some idea what the relativistic case would be, let us consider the quantization in the space ( x , v ) of the relativistic free particle motion. In this case, one makes V ( x ) = 0 on the expression (41) and propose a plane wave solution of the form
ψ ( x , t ) ∼ e i ( k ⋅ x − ω t ) , (46)
on the resulting equation
( 1 + ℏ 2 m 2 c 2 ∇ 2 ) ( − i ℏ ∂ ∂ t + m c 2 ) 2 Ψ = m 2 c 4 Ψ . (47)
Thus, one gets the dispersion relation given by
ω ( k ) = m c 2 ℏ [ 1 1 − ℏ 2 k 2 m 2 c 2 − 1 ] . (48)
See following figure (
The general solution would be the superposition of all the solutions,
Ψ ( x , t ) = ∫ ℜ 3 A ( k ) e i ( k ⋅ x − ω ( k ) t ) d 3 k , (49)
where, due to conservation of probability, the function A ( k ) is such a
∫ ℜ 3 | A ( k ) | 2 d 3 k = 1. (50)
It has been done an extension of the Shrödinger’s quantization approach to the quantization on the space ( x , v ) through the function K ( x , v , t ) which has energy units. Within this approach, the Hamiltonian notion is not needed, and the quantization of conservative systems is the same with this approach and the Hamiltonian approach (in fact, it must be the same whenever the generalized linear momentum is of the form p = m v ). The possibility to include the quantization of mass variation problems and velocity depending problems (dissipation) is clearly stablished. In addition, the quantization of non relativistic interaction of charged particles with electromagnetic field is also stablished.
The author declares no conflicts of interest regarding the publication of this paper.
López, G.V. (2022) About Quantum Mechanics without Hamiltonians. Journal of Modern Physics, 13, 671-681. https://doi.org/10.4236/jmp.2022.135037