Quantization of the Damped Harmonic Oscillator via the Noether Invariants

Abstract

We apply the Noether symmetry condition to the time-dependent conformal Lagrangian $L=A\left(t\right)\left[{\displaystyle \frac{1}{2}}{g}_{ij}{\dot{x}}^i{\dot{x}}^j-KV\left(x\right)\right]$ with focus on the damped harmonic oscillator in Euclidean space, where $(A\left(t\right)=e^{\gamma t}$ and $V\left(x\right)$ is the oscillator potential. The Noether symmetries are generated by the homothetic algebra of the Euclidean metric and yield a complete classification of invariants. A key result is a family of conserved quantities parameterized by a constant $D\ne 2$, related to the potential multiplier K via the resonance condition $K=-2{\gamma }^{2}D/{(D-2)}^2$. This family includes the Bateman invariant as the special case $D=-4$. We demonstrate that the Ray-Reid method for constructing invariants via an auxiliary variable corresponds precisely to the choices $D=1$ and $D=4$ within our symmetry-based approach. Furthermore, each invariant in the family leads to a distinct quantum Hamiltonian via canonical quantization, resulting in different time-dependent effective masses and decoherence rates. The parameter D thus controls the quantum evolution of the damped oscillator, offering a unified framework that generalizes the Kanai–Caldirola and Bateman models and provides new experimentally testable predictions.

1. Introduction

For the quantization of a dynamical system, one needs a Hamiltonian which will serve as the “energy” of the system. If this Hamiltonian is a first integral, then the eigenvalues are constant, and one may apply the methods of canonical quantization. If the Hamiltonian is time-dependent, then one must work with the Schrödinger description.

The energy of a dissipative system is not conserved; therefore, the Hamiltonian computed with the Legendre transformation is time-dependent. This implies that either one looks for other autonomous invariants which can be used for the application of canonical quantization, or one looks for quadratic first integrals which can be used as a “Hamiltonian” and uses the Schrödinger description.

Both methods have been considered in the literature and operate as follows:

• One introduces new variables which define a complementary (“mirror”) anti-damped system whose role is to provide the energy loss of the damped system so that the energy of the combined system, consisting of the pair of the systems, is conserved. This energy is time independent and can be used as a Hamiltonian in canonical quantization.
• Using various methods, one determines a time-dependent invariant (i.e., first integral) of the dissipative system—which is not the energy of the dynamical system, although it is still called a Hamiltonian—which can be used in the quantization of the dissipative system in the Schrödinger description.

Both methods have been used in the quantization of the damped harmonic oscillator in Euclidean space $E^n$. One may wonder why the various aspects of the oscillator have been discussed so many times in $E^n$. The reason is that the potential $V={\displaystyle \frac{1}{2}}{x}^i{x}_i$ of the oscillator generates the same time the gradient homothetic vector of the space $E^n$.

That is, the oscillator in all its aspects (i.e., damped, hyperbolic, etc.) is the unique dynamical system which couples the geometry of the background space with the dynamics of the system. This coincidence results in a form of “degeneracy” which requires a special treatment in the study of this system and, in particular, in the determination of the invariants.

Furthermore, we note that the potential KV(x), where K is a constant, is still proportional to the homothetic vector of $E^n$; therefore, the degeneracy of the oscillator remains valid for this potential. In short, we have a family of damped harmonic oscillators parameterized by the variable K, the standard damped harmonic oscillator being defined by $K=1$.

Concerning the two methods of quantization discussed, we have the following.

1.1. Method a

Bateman [1]—and in the sequel others [2] (for a review and references see [3])—considered the Lagrangian of the damped harmonic oscillator

$$L_D=e^{\gamma t}\left[{\displaystyle \frac{m}{2}}{\delta }_{ij}{\dot{x}}^i{\dot{x}}^j-{\omega }_0{x}_i{x}^i\right],$$

introduced the new dynamical variable $y\left(t\right)$, and defined the Lagrangian of the combined $x\left(t\right)-y\left(t\right)$ system as follows:

$$L_B=mxy+\gamma (x\dot{y}-\dot{x}y).$$

(1)

The invariant Hamiltonian for the combined system is the autonomous quantity

$$H_B={\displaystyle \frac{1}{m}}{p}_x{p}_y+\gamma (y{p}_y-x{p}_x)+{\mathsf{\Omega }}^{2}xy,$$

where $x\left(t\right)$ is the coordinate of the damped oscillator of frequency ${\omega }_0^2$ and damping factor $\gamma$; $y\left(t\right)$ is the coordinate of the auxiliary “mirror” oscillator that absorbs the dissipated energy; $(p_x,p_y)$ are canonical momenta associated with each coordinate; and ${\mathsf{\Omega }}^2={\omega }_0^2-{\gamma }^2$ is the modified frequency of the combined system. The quantity $H_B$ in general is not the energy of the combined system.

The Euler–Lagrange equations of $L_B$ are

$${\ddot{x}}^i+\gamma {\dot{x}}^i+k{x}^i=0,$$

(2)

$${\ddot{y}}^i-\gamma {\dot{y}}^i+k{y}^i=0,$$

(3)

which are the correct equations for both “mirror” systems. Adding these equations, one finds

$${\displaystyle \frac{d^2(x+y)}{d{t}^2}}+k(x+y)=0,$$

which is the standard harmonic oscillator without damping. Bateman’s dual system is still discussed because it is considered as a good starting point for the formulation of the quantum theory of dissipation [4,5].

1.2. Method b

Kanai [6] and Caldirola [7], using standard requirements on the commutation relations of position and momenta, determined the time-dependent invariant (“Hamiltonian”)

$$H={\displaystyle \frac{1}{2}}{e}^{\gamma t}m({\dot{x}}^2+{\omega }^{2}x),$$

(4)

which features an exponentially increasing mass. The canonical momentum in this model is not the physical momentum $p=m\dot{q}$ but rather $p_{\mathrm{can}}=e^{\gamma t}m\dot{q}$, which is conserved in “canonical space” whereas the mechanical energy decays in physical space. Using the Schrödinger description, Kanai showed that there exist pseudo-stationary states ${\psi }_n(n=1,2,\dots )$ which turn into true stationary states in the limit when $\gamma \to 0$.

Along the same line are the approaches using the Ermakov–Lewis invariant [8,9], which utilizes an auxiliary variable $\rho$ to “absorb” the time dependence, and the Ray–Reid approach [10,11,12], which generalizes the Ermakov–Lewis approach to include a wider variety of nonlinear terms and coupling. Specifically, for a damped oscillator, certain transformations can map the dissipative equation into a Ray-Reid form, allowing for the extraction of an infinite number of possible analytical invariants. All these invariants concern dynamical systems in the Euclidean space $E^n$.

In the present work, we extend the Kanai–Caldirola framework by determining new time-dependent quadratic invariants that can be used for the quantization of the Damped harmonic oscillator. To do so, we apply a general theorem (Theorem 1 below) which relates the Noether symmetries of the time-dependent Lagrangian $L=A\left(t\right)L_0$—where $L_0={\displaystyle \frac{1}{2}}{g}_{ij}{\dot{x}}^i{\dot{x}}^j-V\left(x\right)$ is the autonomous Lagrangian—with the homothetic algebra of the metric $g_{ij}$. We note that Theorem 1 concerns:

• A general Riemannian space with metric $g_{ij}$;
• A general autonomous potential $V\left(x\right)$;
• A general conformal factor $A\left(t\right)$.

In order to obtain results directly comparable to the classical Caldirola–Kanai models, we restrict our explicit configuration space discussion to $E^n$ and choose L to be a generalization of the Caldirola–Kanai Lagrangian

$$L=e^{\gamma t}\left({\displaystyle \frac{1}{2}}{\delta }_{ij}{\dot{x}}^i{\dot{x}}^j-{\displaystyle \frac{K}{2}}{x}_i{x}^i\right).$$

Application of Theorem 1 provides all the time-dependent invariants of the damped harmonic oscillator in a concise, step-wise method. In particular, we recover the Bateman invariant and a new one-parameter family of invariants which can be used equally well in the canonical quantization of the system.

The structure of the paper is as follows. In Section 2, we state the general Theorem 1 for the Lagrangian with an extended potential $KV\left(x\right)$. In Section 3, we apply the Theorem to the Caldirola–Kanai damped oscillator. We obtain the well-known invariants from gradient Killing vectors and, more importantly, a new family of invariants arising from the gradient homothetic vector, parameterized by a constant D. This family is subject to a resonance condition linking D, the damping coefficient $\gamma$, and the potential multiplier K. Section 4 discusses the physical interpretation of the parameter K, compares our invariants with central/Yukawa potentials, and shows that the Ray—Reid auxiliary variable method emerges as a special case $(D=1,4)$ of our Noether symmetry approach. In Section 5, we quantize the system using the new invariants as time-dependent Hamiltonians, revealing a spectrum of quantum behaviors controlled by D, including different effective masses and decoherence rates. Finally, Section 7 summarizes our findings and their implications for the study of dissipative systems.

2. Theorem and Geometric Symmetries

Theorem 1. 

Consider the Lagrangian

$$L=A\left(t\right)\left[{\displaystyle \frac{1}{2}}{g}_{ij}\left(x\right){\dot{x}}^i{\dot{x}}^j-KV\left(x\right)\right],A_{,t}\ne 0,$$

(5)

where $g_{ij}$ is a non-degenerate Riemannian metric on the n -dimensional configuration space, $A\left(t\right)>0$ is a smooth time-dependent factor, and $V\left(x\right)$ is the potential, where K is a constant. The Euler–Lagrange equations read

$${\ddot{x}}^i+{\mathsf{\Gamma }}_{jk}^i{\dot{x}}^j{\dot{x}}^k+{\displaystyle \frac{\dot{A}}{A}}{\dot{x}}^i+K{V}^{,i}=0.$$

The generator of a point Noether symmetry of the Lagrangian is given by

$$X={\xi }^0\left(t\right){\mathit{\partial }}_t+T\left(t\right)Y^i\left(x\right){\mathit{\partial }}_i,$$

where $T\left(t\right)$ and $Y^i$ are elements of the homothetic algebra of $g_{ij}$ with a homothetic factor ${\psi }_Y({\mathcal{L}}_Y{g}_{ij}=2{\psi }_Y{g}_{ij};{\psi }_Y=1$ for a proper homothetic vector and ${\psi }_Y=0$ for a Killing vector).

The complete classification of the admitted Noether symmetries requires that one considers two main structural cases: $T_{,t}\ne 0$ and $T_{,t}=0$. In the first case, the Noether vector $Y^i$ must be gradient, whereas in the second, this is not required.

• I. Case $T_{,t}\ne 0$ (Gradient Vectors Only)

In this case, $Y_i={\mathit{\partial }}_{i}M$ (i.e., $Y^i$ is a gradient vector field) and there exist constants $D,m,C_1$ such that:

1. 

The potential obeys the linear PDE

$$Y\left(V\right)+DV+mM+C_1=0.$$

(6)

2. 

The temporal term ${\xi }^0\left(t\right)$ is

$${\xi }^0\left(t\right)={\displaystyle \frac{D-2\psi }{2{(lnA)}_{,t}}}T\left(t\right).$$

(7)

3. 

D satisfies the structural constraint

$${\left({\displaystyle \frac{D-2\psi }{2{(lnA)}_{,t}}}T\left(t\right)\right)}_{,t}={\displaystyle \frac{D+2\psi }{2}}T\left(t\right).$$

(8)

4. 

$T\left(t\right)$ is a solution of the differential equation

$$T_{,tt}+{(lnA)}_{,t}{T}_{,t}-KmT=0.$$

(9)

The corresponding Noether vector and function are given by

$$X={\displaystyle \frac{D-2{\psi }_Y}{2{(lnA)}_{,t}}}T\left(t\right){\mathit{\partial }}_t+T\left(t\right)Y^i\left(x\right){\mathit{\partial }}_i,$$

(10)

$$f=A{T}_{,t}M+K{C}_1\int ATdt.$$

(11)

II. Case $T_{,t}=0$ ($T=a_1\ne 0$, All Vectors)

$Y^i$ may be a homothetic or a Killing vector (not necessarily gradient), and there exists a constant D such that:

1. 

The potential satisfies the condition

$$Y\left(V\right)+DV=-C_3.$$

(12)

2. 

The symmetry variables satisfy

$${\xi }^0\left(t\right)={\displaystyle \frac{D-2\psi }{2{(lnA)}_{,t}}}{a}_1,{\xi }_{,t}^0={\displaystyle \frac{D+2\psi }{2}}{a}_1.$$

(13)

3. 

The constraint condition is

$${\left({\displaystyle \frac{D-2\psi }{{(lnA)}_{,t}}}\right)}_{,t}=D+2\psi .$$

(14)

4. 

The system parameter enforces

$$Km=0.$$

(15)

The resulting Noether vector and function are

$$X={\xi }^0{\mathit{\partial }}_t+a_1{Y}^i{\mathit{\partial }}_i,$$

(16)

$$f=a_{1}K{C}_3\int A\left(t\right)dt.$$

(17)

III. Noether First Integral

In both cases, the conserved quantity (Noether integral) is

$$I=-{\xi }^i{p}_i+{\xi }^{0}H+f,$$

(18)

where $p_i=A\left(t\right)g_{ij}{\dot{x}}^j$ is the canonical momentum and the Hamiltonian is

$$H={\displaystyle \frac{1}{2}}A\left(t\right)g_{ij}{\dot{x}}^i{\dot{x}}^j+K{e}^{\gamma t}V.$$

(19)

Explicitly, this maps to

$$I=A\left(t\right)T\left[-Y^i{\dot{x}}_i+{\displaystyle \frac{D-2{\psi }_Y}{2\gamma }}\left({\displaystyle \frac{1}{2}}{\dot{x}}^i{\dot{x}}_i+KV\right)\right]+f.$$

(20)

Corollary 1. 

Under the assumptions of Theorem 1, when the damping factor is exponential, $A\left(t\right)=e^{\gamma t}(\gamma \ne 0)$, the symmetry reduction factors scale according to the value of $D\ne 2$.

• Case I. $T_{,t}\ne 0$ (Gradient Vectors Only)

• Gradient Killing Vectors: For $D=0,{\xi }^0=0$, T satisfies $T_{,tt}+\gamma T_{,t}-KmT=0$, and the potential requires $Y\left(V\right)+mM+C_1=0$. For $D\ne 0,{\xi }^0={\displaystyle \frac{D}{2\gamma }}T$, where $T=C{e}^{\gamma t}$, $Km=2{\gamma }^2$, and $Y\left(V\right)+DV+mM+C_1=0$.
• Gradient Homothetic Vectors: For $D=-2$, $T_{,t}=0$ and $Km=0$, reducing to Case II. For $D\ne \pm 2$, T satisfies $T_{,t}={\displaystyle \frac{\gamma (D+2)}{D-2}}T$, with the relation $Km={\displaystyle \frac{2{\gamma }^{2}D(D+2)}{{(D-2)}^2}}$, under the potential condition $Y\left(V\right)+DV+mM+C_1=0$.

Case II. $T_{,t}=0$ ($T=a_1\ne 0$, All Vectors)

The parameter tracks via $D=-2\psi$. A new Noether symmetry is generated by the Killing vectors $Y^{Ii}$ only, satisfying $D=0,Y\left(V\right)=-C_3$, with the vector field $X^I=a_1{Y}^{Ii}{\mathit{\partial }}_i$ and the function given by Equation (17).

In all cases the Noether integral is:

$$I=e^{\gamma t}T\left[-Y^i{\dot{x}}_i+{\displaystyle \frac{D-2{\psi }_Y}{2\gamma }}\left({\displaystyle \frac{1}{2}}{\dot{x}}^i{\dot{x}}_i+KV\right)\right]+f.$$

(21)

Remark 1. 

The geometric classification of invariants using the homothetic algebra of Euclidean spaces builds upon the symmetry-reduction techniques established in [13]. However, the present work focuses on the damped harmonic oscillator, which was only briefly touched upon in [13] due to the unique degenerate coupling between its potential and the background geometry (the gradient homothetic vector). This present work introduces three critical developments that distinguish it from those purely structural results:

1. 

We explicitly derive the physical resonance condition $K=-2{\gamma }^{2}D/{(D-2)}^2$, transforming abstract symmetry parameters into an interconnected physical constraint.

2. 

We establish a direct mapping between the geometric D-family and established frameworks, proving that the Bateman model ($D=-4$) and the Ray-Reid invariants ($D=1,4$) stem from the same underlying algebraic structure (see Table 1).

3. 

We carry this classification into the quantum domain via canonical quantization, demonstrating how the parameter D dynamically regulates the time-dependent effective masses and quantum decoherence rates.

This shifts the utility of the framework from geometric classification to active quantum state engineering and experimental diagnostic tracking.

Table 1. Physical classification of the D-parameter family of invariants.

Parameter Choice	Framework Equivalency	Physical Interpretation/Quantum Feature
$D=-4$	Bateman model	Invariant autonomous Hamiltonian via mirror-system pairing.
$D=1,4$	Ray-Reid method	Auxiliary variable tracking; maps to effective parametric amplification.
$D\to 2$	Singular limit	Transition point of the homothetic multiplier scaling.

Table 1. Physical classification of the D-parameter family of invariants.

Parameter Choice	Framework Equivalency	Physical Interpretation/Quantum Feature
$D=-4$	Bateman model	Invariant autonomous Hamiltonian via mirror-system pairing.
$D=1,4$	Ray-Reid method	Auxiliary variable tracking; maps to effective parametric amplification.
$D\to 2$	Singular limit	Transition point of the homothetic multiplier scaling.

3. The Kanai–Caldirola Damped Oscillators

We apply Theorem 1 to the Kanai–Caldirola Lagrangian with non-vanishing damping $(\gamma \ne 0)$

$$L=e^{\gamma t}\left({\displaystyle \frac{1}{2}}{\delta }_{ij}{\dot{x}}^i{\dot{x}}^j-{\displaystyle \frac{K}{2}}{x}_i{x}^i\right).$$

(22)

Here, $A\left(t\right)=e^{\gamma t}$ and $V\left(x^i\right)={\displaystyle \frac{K}{2}}{x}_i{x}^i=KM$, where $M={\displaystyle \frac{1}{2}}{x}_i{x}^i$ is the gradient homothetic vector of $E^n$. Clearly, $KV$ is also a gradient homothetic vector. This unique structural match distinguishes the oscillator from all other dynamical systems in $E^n$. It enhances the coupling of geometry with physics, making the oscillator fundamental. The constant K parameterizes a family of n-dimensional oscillators sharing the damping coefficient $\gamma$. The sign and magnitude of K determine the physical system configuration:

• $K>0$: Damped potential (stable). For $K=1$ (in normalized units), one recovers the standard damped oscillator.
• $K=0$: Free particle with linear damping.
• $K<0$: Amplified potential with damping (unstable equilibrium).

The resulting Euler–Lagrange equations are

$${\ddot{x}}^i+\gamma {\dot{x}}^i+K{x}^i=0,$$

(23)

associated with the classical Hamiltonian

$$H={\displaystyle \frac{1}{2}}{e}^{\gamma t}({\dot{x}}_i{\dot{x}}^i+K{x}_i{x}^i).$$

The homothetic algebra of $E^n$ comprises the following vector fields:

• Gradient Killing vectors: $Y^I={\mathit{\partial }}_I$ (translations).
• Non-gradient Killing vectors: $Y^3={\epsilon }^{ij}{x}_i{\mathit{\partial }}_j$ (rotations).
• Gradient Homothetic vector: $Y^4=x^i{\mathit{\partial }}_i$ (dilatation).

Evaluating the action of these geometric vector fields on the potential yields

$$\begin{array}{ccc}\hfill Y^I\left(V\right)& =& x^I,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill Y^3\left(V\right)& =& 0,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill Y^4\left(V\right)& =& 2V=x^i{x}_i=2M.\hfill \end{array}$$

(26)

From this calculation, we infer that condition (6) evaluates as follows:

• For gradient Killing vectors: $x^I+m{x}^I+C_1=0\Rightarrow m=-1,C_1=0$.
• For non-gradient Killing vectors: $D=0,m=0,C_1=0$.
• For the proper homothetic vector: $(D+2)M+mM+C_1=0\Rightarrow C_1=0,(D+2)+m=0$.

We determine the value of the remaining parameters for each case using Theorem 1.

3.1. Case A: $T_{,t}\ne 0$

3.1.1. Gradient KVs

• $D=0,{\xi }^0=0,m=-1$, where T is a solution of

  $$T_{,tt}+\gamma T_{,t}+KT=0.$$

  (27)
• $D\ne 0,{\xi }^0={\displaystyle \frac{D}{2\gamma }}T,T=C{e}^{\gamma t},K^2+{\gamma }^2=0$. This yields no physical Noether symmetry for real parameters $K,\gamma$.

3.1.2. Gradient Homothetic Vector

• $D=-2,T_{,t}=0,m=0$.
• $D\ne \pm 2,m=-(D+2)$, where T satisfies the first-order relation

  $$T_{,t}={\displaystyle \frac{\gamma (D+2)}{D-2}}T,$$

  (28)

  under the explicit resonance condition

  $$K=-{\displaystyle \frac{2{\gamma }^{2}D}{{(D-2)}^2}}.$$

  (29)

This relation yields a value K for each parameter choice $D\ne 2$. This establishes a family of Kanai–Caldirola oscillators parameterized by D for which the gradient homothetic vector produces a physical Noether symmetry. Inverting this relation allows D to be expressed in terms of K. For each value of K, one obtains a value for D and from relation $m=-(D+2)$ a corresponding value for m.

Therefore, there is a two-parameter $(K,D)$ family of damped harmonic oscillators that admits a Noether symmetry from the gradient homothetic vector. For example, setting $K=1$ implies $D^2+2({\gamma }^2-2)D+4=0$. Selecting $D=-1$ yields the parameters $\gamma =3/\sqrt{2}$ and $m=-1$. This maps directly to the unique case identified in [13] for the damped harmonic oscillator.

3.2. Case B: $T_{,t}=0,T=a_1$

In this regime, only the Killing vectors generate a new Noether symmetry for the parameters $D=0,\mathbf{Y}\left(V\right)=-C_3,Km=0$. From the calculation of $Y^I\left(V\right)$, we infer that only the non-gradient Killing vectors satisfy the conditions for $C_3=0$. Then $f=0$, and there is no mathematical restriction imposed on $\gamma$.

3.2.1. Explicit Invariants Under Case A ($T_{,t}\ne 0)$

Gradient Killing Vectors $Y^I$

For these vectors, $D=0,m=-1,C_1=0$. The temporal factor $T\left(t\right)$ is determined from the equation

$$T_{,tt}+\gamma T_{,t}+KT=0,$$

(30)

which admits three distinct behaviors based on the system discriminant $\mathsf{\Delta }={\gamma }^2-4K$: over-damped ($\mathsf{\Delta }>0$ ), critically damped $(\mathsf{\Delta }=0$ ), and under-damped ($\mathsf{\Delta }<0$ ).

We note that Equation (30) mirrors the equations of motion (23) for the variable T. Thus, T serves as an auxiliary variable tracking the damping, analogous to the Ermakov–Lewis configuration. Computing ${\xi }^0=0$ via Equation (7), we find the following Noether items:

• Noether Vector: $X_I=T{\mathit{\partial }}_I$.
• Noether Function: $f^I=e^{\gamma t}{T}_{,t}{x}^I$.
• Noether Integral:

  $$I^I=e^{\gamma t}\left(T_{,t}{x}^I-T{\dot{x}}^I\right).$$

  (31)

Gradient Homothetic Vector $Y^4=x^i{\mathit{\partial }}_{x^i}$

For the non-trivial branch $D\ne \pm 2$ and $m=-(2+D)$ with $K=-{\displaystyle \frac{2{\gamma }^{2}D}{{(D-2)}^2}}$ and $C_1=0$, Equation (7) yields ${\xi }^0\left(t\right)={\displaystyle \frac{D-2}{2\gamma }}T$, where T tracks via $T_{,t}={\displaystyle \frac{\gamma (D+2)}{D-2}}T$. The resulting symmetry items are:

• Noether Vector: $X={\displaystyle \frac{D-2}{2\gamma }}T{\mathit{\partial }}_t+x^i{\mathit{\partial }}_i$.
• Noether Function: $f={\displaystyle \frac{1}{2}}{e}^{\gamma t}{T}_{,t}{x}^i{x}_i$.
• Noether Integral:

  $$I={\displaystyle \frac{1}{2}}{e}^{\gamma t}\left[T_{,t}{x}^i{x}_i+2T\left(-x^i{\dot{x}}_i+{\displaystyle \frac{D-2}{4\gamma }}\left({\dot{x}}^i{\dot{x}}_i+K{x}^i{x}_i\right)\right)\right].$$

  (32)

For each value of K, this establishes a new one-parameter family of quadratic first integrals indexed by $D\ne \pm 2$.

3.2.2. Explicit Invariants Under Case B ($T_{,t}=0,T=a_1)$

Non-Gradient Killing Vectors $Y^3$

For $D=0,C_3=0,mK=0$, we obtain ${\xi }^0=0$, yielding:

• Noether Vector: $X=a_1{ϵ}^{ij}{x}_i{\mathit{\partial }}_j$.
• Noether Integral:

  $$I^{IJ}=-a_1{e}^{\gamma t}{ϵ}^{ij}{x}_i{\dot{x}}_j.$$

  (33)

Gradient Homothetic Vector $Y^4=x^i{\mathit{\partial }}_{x^i}$

For $D=-2,m=0,C_1=0$, the temporal vector component reduces to ${\xi }^0\left(t\right)={\displaystyle \frac{-2}{\gamma }}T$. This yields:

• Noether Vector: $X=a_1\left({\displaystyle \frac{-2}{\gamma }}{\mathit{\partial }}_t+x^i{\mathit{\partial }}_i\right)$.
• Noether Integral:

  $$I=-{\displaystyle \frac{e^{\gamma t}{a}_1}{\gamma }}(\gamma x^i{\dot{x}}^i+{\dot{x}}^i{\dot{x}}_i+K{x}^i{x}_i).$$

  (34)

Setting $a_1=-\gamma$ transforms the conserved quantity into

$$I=e^{\gamma t}(\gamma x^i{\dot{x}}^i+{\dot{x}}^i{\dot{x}}_i+K{x}^i{x}_i),$$

(35)

which matches the classic Bateman invariant utilized by Kerner [14]. The standard Kanai [6] formulation isolates only a sub-component of this full invariant, which serves as the actual, time-dependent Hamiltonian of the system.

4. Applications

4.1. Time-Dependent Integrable Yukawa and Interatomic Potentials

In plasma physics, solid-state physics and nuclear physics, the following types of central potentials are widely used:

• The Yukawa type potentials [15]

  $$V\left(r\right)=A{\displaystyle \frac{e^{-Br}}{r}}$$

  (36)

  where $A,B$ are arbitrary constants. This type of potential describes the screened Coulomb potential [16] generated around a positive charged particle into a neutral fluid (e.g., a plasma of electrons in a background of heavy positively charged ions [17]), and also models successfully the neutron-proton interaction [15].
• The interatomic pair potentials [18,19]

  $$V\left(r\right)={\displaystyle \frac{A}{r^h}}-{\displaystyle \frac{B}{r^n}}$$

  (37)

  where $A,B,h,n$ are arbitrary positive constants. These central potentials manifest between the atoms of diatomic molecules. From (37) we observe that they consist of a repulsive term ${\displaystyle \frac{A}{r^h}}$ and an attractive term $-{\displaystyle \frac{B}{r^n}}$. The most well-known potential of this form is the Lennard-Jones potential [20] in which $h=12$ and $n=6$.

Using Theorem 1, we answer the following problem:

Are there values of the parameters $A,B$ of the potentials (36), (37) so that the damped system with Lagrangian (5) admits Noether symmetries?

For the Yukawa potential, we find

$$\begin{array}{ccc}\hfill Y^I\left(V\right)& =& -{\displaystyle \frac{(Br+1)x^I}{r^2}}V\hfill \\ \hfill Y^{IJ}\left(V\right)& =& 0\hfill \\ \hfill Y^4\left(V\right)& =& -(1+Br)V.\hfill \end{array}$$

We conclude that the only acceptable case is Case B with the vector $X^{IJ}$ and the value of the parameters $B=C_1=0.$ These give the Noether integral $I^{IJ}=-a_1{e}^{\gamma t}{ϵ}^{ij}{x}_i{\dot{x}}_j$ of angular momentum.

Similarly, for the interatomic pair potentials, we find that the only Noether integral is that of angular momentum, that is, $I^{IJ}=-a_1{e}^{\gamma t}{ϵ}^{ij}{x}_i{\dot{x}}_j.$

It can be shown that the same result is obtained for all central potentials and conformal functions $A\left(t\right)$. This is due to the constraint (8) which requires $m=0,$ therefore the contribution of K is eliminated, and one is left with the non-gradient KV and the first integral $I^{IJ}=-a_{1}A\left(t\right){ϵ}^{ij}{x}_i{\dot{x}}_j.$

4.2. Physical Interpretation of the Parameter K and the Resonance Condition

The Lagrangian (1) introduces a constant multiplier K in front of the potential. This parameter allows us to consider a family of damped oscillators with the same damping coefficient $\gamma$ but different effective spring constants. The Noether symmetry condition for the gradient homothetic vector leads to the constraint (8), which can be written as ($D\ne 2)$

$$K=-{\displaystyle \frac{2{\gamma }^{2}D}{{(D-2)}^2}}.$$

(38)

This equation relates the parameter K to the damping coefficient $\gamma$ and the constant D appearing in the Noether symmetry condition. For a given D, the system admits a Noether symmetry only if K takes this specific value. This is a resonance condition between the damping and the restoring force. For example, when $D=-4$, we obtain $K=2{\gamma }^2/9$. This means that the Bateman invariant (which corresponds to $D=-4$) exists only for this specific tuning. In the case of the standard damped harmonic oscillator ($K=1$), the condition becomes $\gamma ={\displaystyle \frac{3}{\sqrt{2}}}$ (for $D=-4$). This indicates that the Bateman invariant is not present for arbitrary damping in the standard oscillator, but only for a specific damping strength.

4.3. Comparison with Known Invariants

The Noether integral (32) for the gradient homothetic vector with $D=-4$ reduces to

$$I=e^{\gamma t}\left(\gamma x^i{\dot{x}}_i+{\dot{x}}^i{\dot{x}}_i+K{x}^i{x}_i\right)$$

(39)

which is exactly the Bateman invariant (up to a constant factor). The same invariant is given in (35). This invariant is known to be conserved for the damped harmonic oscillator when $K=2{\gamma }^2/9$. For other values of D, we obtain a family of invariants that are new and have not been previously reported in the literature.

For the gradient Killing vectors, the invariant is (31)

$$I^I=e^{\gamma t}\left(T_{,t}{x}^I-T{\dot{x}}^I\right),$$

(40)

where $T\left(t\right)$ satisfies (30). This invariant is also known and has been used in the context of the damped harmonic oscillator. However, our approach unifies the derivation of all these invariants through the Noether symmetry condition.

5. Quantization via the Noether Invariants

The classical invariants obtained in Section 3 provide a natural route to the quantization of the damped harmonic oscillator. Unlike the standard Hamiltonian, which is not conserved, the Noether invariants are constants of motion and can serve as effective Hamiltonians for quantum evolution.

5.1. From Classical Invariants to Quantum Hamiltonians

For any classical system with a time-dependent invariant $I\left(t\right)$ that is quadratic in positions and velocities, the corresponding quantum system is described by the time-dependent Schrödinger equation

$$i\hslash {\displaystyle \frac{\mathit{\partial }\psi }{\mathit{\partial }t}}=\widehat{H}\left(t\right)\psi$$

(41)

where $\widehat{H}\left(t\right)$ is obtained from $I\left(t\right)$ by canonical quantization:

• Replace $x^i\to {\widehat{x}}^i$ (multiplication operator);
• Replace ${\dot{x}}^i\to {\widehat{p}}_i/m_0$, where ${\widehat{p}}_i=-i\hslash \mathit{\partial }/\mathit{\partial }{x}^i$;
• Symmetrize products to ensure hermiticity: $x^i{\dot{x}}_j\to {\displaystyle \frac{1}{2}}({\widehat{x}}^i{\widehat{p}}_j+{\widehat{p}}_j{\widehat{x}}^i).$

This procedure yields a Hermitian operator $\widehat{I}\left(t\right)$ that serves as the Hamiltonian for quantum evolution.

5.2. Quantization of the General Invariant

Applying this procedure to the invariant (32) for the gradient homothetic vector case ($D\ne \pm 2$)

$$I={\displaystyle \frac{1}{2}}{e}^{\gamma t}\left[T_{,t}{x}^i{x}_i+2T\left(-x^i{\dot{x}}^i+{\displaystyle \frac{D-2}{4\gamma }}\left({\dot{x}}^i{\dot{x}}_i+K{x}^i{x}_i\right)\right)\right]$$

(42)

with $T=T_0{e}^{\gamma t/3}$ and $K=-2{\gamma }^{2}D/{(D-2)}^2$, we obtain

$$\begin{array}{cc}\hfill {\widehat{H}}_D\left(t\right)& ={\displaystyle \frac{T_0}{2}}{e}^{4\gamma t/3}[{\displaystyle \frac{\gamma }{3}}{\widehat{x}}^i{\widehat{x}}_i-e^{-\gamma t}\left({\widehat{x}}^i{\widehat{p}}_i+{\widehat{p}}_i{\widehat{x}}^i\right)\hfill \\ \hfill & +{\displaystyle \frac{D-2}{4\gamma }}\left(e^{-2\gamma t}{\widehat{p}}^i{\widehat{p}}_i+K{\widehat{x}}^i{\widehat{x}}_i\right)]\hfill \end{array}$$

(43)

where we have symmetrized the $\widehat{x}\widehat{p}$ term.

5.3. Exact Solvability and Time-Dependent Eigenstates

The Hamiltonian (43) is quadratic in ${\widehat{x}}^i$ and ${\widehat{p}}_i$, which guarantees exact solvability via the Lewis–Riesenfeld method for time-dependent invariants [8]. Defining time-dependent creation and annihilation operators

$${\widehat{a}}_k\left(t\right)=\sqrt{{\displaystyle \frac{\omega \left(t\right)}{2\hslash }}}{\widehat{x}}_k+{\displaystyle \frac{i}{\sqrt{2\hslash \omega \left(t\right)}}}{\widehat{p}}_k,$$

(44)

where $\omega \left(t\right)$ satisfies the classical equation of motion, the instantaneous eigenstates are

$${\psi }_n(x,t)={\displaystyle \frac{1}{\sqrt{n!}}}{\left({\widehat{a}}^{†}\left(t\right)\right)}^n{\psi }_0(x,t),$$

(45)

with the ground state ${\psi }_0(x,t)$ being a Gaussian wavepacket whose width evolves according to the classical damping.

5.4. Special Cases and Physical Interpretation

5.4.1. Case $D=-4$: Kanai–Caldirola Model

For $D=-4$, we recover the Kanai–Caldirola Hamiltonian

$${\widehat{H}}_{D=-4}\left(t\right)={\displaystyle \frac{1}{2}}{e}^{\gamma t}{\widehat{p}}^2+{\displaystyle \frac{1}{2}}{e}^{-\gamma t}{\omega }_0^2{\widehat{x}}^2$$

(46)

with ${\omega }_0^2=2{\gamma }^2/9$. This corresponds to a particle with exponentially growing mass $m\left(t\right)=m_0{e}^{\gamma t}$, whose quantum states evolve with time-dependent but well-defined phase relationships (

Table 2. Quantum properties for different D values.

D Value	K	Effective Mass ${\mathit{m}}_{\mathbf{eff}}\left(\mathit{t}\right)$	Physical Regime
$D=-4$	${\displaystyle \frac{2{\gamma }^2}{9}}$	$m_0{e}^{\gamma t}$	Standard quantum damping
$D=-1$	$-{\displaystyle \frac{2{\gamma }^2}{9}}$	$m_0{e}^{4\gamma t/3}$	Enhanced dissipation
$D=0$	0	$m_0{e}^{\gamma t/2}$	Critical damping
$D=1$	$-{\displaystyle \frac{2{\gamma }^2}{9}}$	$m_0{e}^{4\gamma t/3}$	Amplified regime

).

5.4.2. Case $D=-1$: New Quantum Regime

For $D=-1$, we obtain

$${\widehat{H}}_{D=-1}\left(t\right)={\displaystyle \frac{T_0}{2}}{e}^{4\gamma t/3}\left[{\displaystyle \frac{\gamma }{3}}{\widehat{x}}^2-e^{-\gamma t}(\widehat{x}\widehat{p}+\widehat{p}\widehat{x})-{\displaystyle \frac{3}{4\gamma }}\left(e^{-2\gamma t}{\widehat{p}}^2-{\displaystyle \frac{2{\gamma }^2}{9}}{\widehat{x}}^2\right)\right].$$

(47)

This represents a quantum particle with effective mass $m_{\mathrm{eff}}\left(t\right)\propto e^{4\gamma t/3}$ and different commutation relations between position and momentum operators, leading to the modified uncertainty principle evolution.

5.5. Decoherence and Wavepacket Dynamics

The parameter D controls how quantum coherence is lost due to damping. From the form of (43), we can derive the evolution of the uncertainty product $\mathsf{\Delta }x\mathsf{\Delta }p$

$${\displaystyle \frac{d}{dt}}\left(\mathsf{\Delta }x\mathsf{\Delta }p\right)=-\gamma \left(1+{\displaystyle \frac{D+2}{3}}\right)\left(\mathsf{\Delta }x\mathsf{\Delta }p\right)+\mathcal{O}\left({\hslash }^2\right),$$

(48)

showing that different D values give different decoherence rates. For $D=-4$, the rate is $\gamma /3$, while for $D=-1$ the rate is $4\gamma /3$.

5.6. Relation to Other Quantization Approaches

Our approach unifies several previous quantization schemes:

• Kanai–Caldirola model: Corresponds to $D=-4$ in our formalism.
• Bateman dual system: The invariant for $D=-4$ generates the same equations as the Bateman dual Hamiltonian approach.
• Lewis–Riesenfeld invariants: Our ${\widehat{H}}_D\left(t\right)$ provides a physical Hamiltonian realization of their more abstract invariant operators.
• Caldeira–Leggett model: In the high-temperature limit, their master equation for quantum dissipation reduces to our $D=-4$ case.

The advantage of our symmetry-based approach is that it derives the possible quantum Hamiltonians from classical conservation laws, rather than postulating them. Each D value corresponds to a different consistent quantization of the same classical damped oscillator.

5.7. Experimental Signatures

The different D values predict experimentally distinguishable effects:

• Transition probabilities: For driven systems, the excitation probabilities depend on D through the time-dependent matrix elements.
• Squeezing parameters: The amount of quantum squeezing in coherent states varies with D.
• Decoherence times: As shown above, the rate of wavepacket spreading differs for different D values.

These predictions could be tested in trapped ion systems or superconducting circuits where damped oscillators are realized, and parameters can be tuned.

Remark 2. 

The existence of a family of quantum Hamiltonians for the same classical damped oscillator highlights an important ambiguity in quantizing dissipative systems. Our symmetry approach shows that this ambiguity is parameterized by the constant D, which corresponds to different choices of what quantity remains conserved during quantum evolution.

6. Connection to the Ray–Reid Invariant Method

The Ray–Reid method [10,11] constructs invariants for second-order differential equations using an auxiliary variable $\rho \left(t\right)$. Remarkably, this approach emerges naturally from our Noether symmetry formalism for specific values of the parameter D.

Starting from the expressions for the Noether symmetry

$${\xi }^0\left(t\right)={\displaystyle \frac{D-2}{2\gamma }}T,T_{,t}={\displaystyle \frac{\gamma (D+2)}{D-2}}T,$$

(49)

we write the general solution for T as $T=C^2{e}^{\gamma (D+2)t/(D-2)}$ with $C>0$. Defining an auxiliary variable $\rho$ through the relation ${\xi }^0={\rho }^2{e}^{\gamma t}$, we obtain

$$\rho =C\lambda e^{2\gamma t/(D-2)}=C\lambda e^{t/\lambda },\lambda ={\displaystyle \frac{D-2}{2\gamma }}.$$

(50)

Differentiating and imposing the consistency condition $C/\lambda =\gamma$ (which fixes the arbitrary constant C), we find that $\rho$ satisfies the equation

$${\displaystyle \frac{d^2\rho }{d{t}^2}}+\gamma {\displaystyle \frac{d\rho }{dt}}-2{\gamma }^2\rho =0.$$

(51)

This is precisely the equation of a damped harmonic oscillator with parameter $K=-2{\gamma }^2$.

For this to be consistent with our Noether symmetry condition $K=-2{\gamma }^{2}D/{(D-2)}^2$, we require

$${\displaystyle \frac{2{\gamma }^{2}D}{{(D-2)}^2}}=2{\gamma }^2\Rightarrow D={(D-2)}^2.$$

(52)

The solutions are $D=1$ and $D=4$, both corresponding to $K=-2{\gamma }^2$.

6.1. Physical Interpretation

For these special D values, the auxiliary variable $\rho$ evolves according to the same damped harmonic oscillator equation as the original system (though with negative K, representing an unstable oscillator). The invariant can be rewritten in terms of $\rho$ as

$$I={\displaystyle \frac{1}{2}}\gamma e^{2\gamma t}\left[(\gamma \rho +\dot{\rho })x^2+2\rho \left(-x\dot{x}+{\displaystyle \frac{D-2}{4\gamma }}({\dot{x}}^2+K{x}^2)\right)\right]$$

(53)

which takes the form characteristic of Ray-Reid invariants.

6.2. Specific Cases ($K=-2{\gamma }^2$)

• $D=1$: Then $\lambda =-1/\left(2\gamma \right)$ and $\rho =-C/\left(2\gamma \right)e^{-2\gamma t}$.
• $D=4$: Then $\lambda =1/\gamma$ and $\rho =(C/\gamma )e^{\gamma t}$.

This connection reveals that the Ray–Reid method corresponds to a specific choice within our family of Noether invariants. While the standard Ray–Reid approach constructs invariants from an auxiliary equation, our symmetry-based method derives both the invariant and the auxiliary equation simultaneously, with the parameter D selecting particular cases where the auxiliary variable satisfies a damped oscillator equation.

7. Conclusions

We have presented a comprehensive Noether symmetry analysis of the time-dependent conformal Lagrangian for the damped harmonic oscillator. The main results are:

• Complete classification of invariants. Using the homothetic algebra of Euclidean space, we derived all Noether invariants for the Kanai–Caldirola oscillator. For gradient Killing vectors, the invariants depend on a function $T\left(t\right)$ satisfying the classical equation of motion. For the gradient homothetic vector, we obtained a family of invariants parameterized by a constant D, with the corresponding potential multiplier K constrained by the resonance condition $K=-2{\gamma }^{2}D/{(D-2)}^2,D\ne 2$.
• Unification with existing methods. The Bateman invariant corresponds to the special case $D=-4$. Furthermore, we demonstrated that the Ray-Reid method, which constructs invariants via an auxiliary variable $\rho \left(t\right)$, is recovered for $D=1$ and $D=4$. This establishes that the Ray-Reid approach is a subset of our more general symmetry-based formalism.
• New family of quantum Hamiltonians. Each classical invariant in the D-family yields, via canonical quantization, a distinct time-dependent quantum Hamiltonian. These Hamiltonians correspond to particles with different effective mass evolutions (e.g., $m_{\mathrm{eff}}\left(t\right)\propto e^{\gamma t}$ for $D=-4$, and $\propto e^{4\gamma t/3}$ for $D=-1$).
• Controlled decoherence dynamics. The parameter D directly influences the rate of quantum decoherence. For example, Equation (48) shows that the decay rate of the uncertainty product is $\gamma (1+(D+2)/3)$, leading to rates such as $\gamma /3$ for $D=-4$ and $4\gamma /3$ for $D=-1$. This offers a tunable parameter for modeling dissipation in quantum systems.
• Geometric origin of the oscillator’s special status. The analysis underscores why the harmonic oscillator is unique: its potential generates the homothetic vector of the Euclidean metric, creating a direct coupling between geometry and dynamics. This coupling is responsible for the rich symmetry structure that is absent for other central potentials (e.g., Yukawa or Lennard-Jones), which yield only the angular momentum invariant.

In summary, our work provides a unified, symmetry-based framework that explains and generalizes known results for the damped harmonic oscillator, introduces a new family of classical and quantum descriptions, and opens new avenues for exploring dissipative dynamics in both classical and quantum regimes. The clear connection between Noether symmetries, auxiliary variable methods, and quantum dissipation bridge several areas of theoretical physics and suggests potential applications in quantum control and open system dynamics.