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TL; DR

• Mathematically the two equations do not belong to the same type
• There is a lot of specific physical content associated with the Schrödinger equation

In purely mathematical terms, the second order partial differential equations are classified into elliptic, parabolic and hyperbolic equations. The most common representatives of these three classes occurring in physics are

• Laplace equation $$\nabla^2 u(x,y,z) = 0$$
• Diffusion equation $$D\nabla^2 u(x,y,z,t) = \frac{\partial u(x,y,z,t)}{\partial t}$$
• Wave equation $$\nabla^2 u(x,y,z,t) = \frac{1}{v^2}\frac{\partial^2 u(x,y,z,t)}{\partial t^2}$$

Although the there names originate from particular applications, they are frequently used to denote particular mathematical type of equation. As yous ee the mathematical classification here is based on the type of derivatives involved (parabolic/diffusion equation ahs a first order derivative) and the sign of the second order derivatives (Laplace equation ahs all the derivatives of the same sign, whereas the wave equation has one derivative with a different sign.)

Then there are even more domain-specific names, which typically imply certain type of coefficients and/or certain type of inhomogeneous terms. Thus, Poisson equation is an inhomogeneous Laplace equation, whereas Schrödinger equation is a Diffusion equation, with a complex diffusion coefficient, often with a term with zero derivative (the potential term) and potentially also the first derivative term (in magnetic field). Moreover, in some settings Schrödinger equation may contain higher order derivatives, be formulated for multicomponent functions (e.g., in presence of spin), and even for descrete rather than continuous functions (although continuous in time). In other words, it is mathematically and physically very different from the wave equation.

TL; DR

• Mathematically the two equations do not belong to the same type
• There is a lot of specific physical content associated with the Schrödinger equation

In purely mathematical terms, the second order partial differential equations are classified into elliptic, parabolic and hyperbolic equations. The most common representatives of these three classes occurring in physics are

• Laplace equation $$\nabla^2 u(x,y,z) = 0$$
• Diffusion equation $$D\nabla^2 u(x,y,z,t) = \frac{\partial u(x,y,z,t)}{\partial t}$$
• Wave equation $$\nabla^2 u(x,y,z,t) = \frac{1}{v^2}\frac{\partial^2 u(x,y,z,t)}{\partial t^2}$$

Although the there names originate from particular applications, they are frequently used to denote particular mathematical type of equation. As you see the mathematical classification here is based on the type of derivatives involved (parabolic/diffusion equation has a first order derivative) and the sign of the second order derivatives (Laplace equation has all the derivatives of the same sign, whereas the wave equation has one derivative with a different sign.)

Then there are even more domain-specific names, which typically imply certain type of coefficients and/or certain type of inhomogeneous terms. Thus, Poisson equation is an inhomogeneous Laplace equation, whereas Schrödinger equation is a Diffusion equation, with a complex diffusion coefficient, often with a term with zero derivative (the potential term) and potentially also the first derivative term (in magnetic field). Moreover, in some settings Schrödinger equation may contain higher order derivatives, be formulated for multicomponent functions (e.g., in presence of spin), and even for descrete rather than continuous functions (although continuous in time). In other words, it is mathematically and physically very different from the wave equation.

In purely mathematical terms, the second order partial differential equations are classified into elliptic, parabolic and hyperbolic equations. The most common representatives of these three classes occurring in physics are

• Laplace equation $$\nabla^2 u(x,y,z) = 0$$
• Diffusion equation $$D\nabla^2 u(x,y,z,t) = \frac{\partial u(x,y,z,t)}{\partial t}$$
• Wave equation $$\nabla^2 u(x,y,z,t) = \frac{1}{v^2}\frac{\partial^2 u(x,y,z,t)}{\partial t^2}$$

Although the there names originate from particular applications, they are frequently used to denote particular mathematical type of equation. As yous ee the mathematical classification here is based on the type of derivatives involved (parabolic/diffusion equation ahs a first order derivative) and the sign of the second order derivatives (Laplace equation ahs all the derivatives of the same sign, whereas the wave equation has one derivative with a different sign.)

Then there are even more domain-specific names, which typically imply certain type of coefficients and/or certain type of inhomogeneous terms. Thus, Poisson equation is an inhomogeneous Laplace equation, whereas Schrödinger equation is a Diffusion equation, with a complex diffusion coefficient, often with a term with zero derivative (the potential term) and potentially also the first derivative term (in magnetic field). Moreover, in some settings Schrödinger equation may contain higher order derivatives, be formulated for multicomponent functions (e.g., in presence of spin), and even for descrete rather than continuous functions (although continuous in time). In other words, it is mathematically and physically very different from the wave equation.

TL; DR

• Mathematically the two equations do not belong to the same type
• There is a lot of specific physical content associated with the Schrödinger equation

In purely mathematical terms, the second order partial differential equations are classified into elliptic, parabolic and hyperbolic equations. The most common representatives of these three classes occurring in physics are

• Laplace equation $$\nabla^2 u(x,y,z) = 0$$
• Diffusion equation $$D\nabla^2 u(x,y,z,t) = \frac{\partial u(x,y,z,t)}{\partial t}$$
• Wave equation $$\nabla^2 u(x,y,z,t) = \frac{1}{v^2}\frac{\partial^2 u(x,y,z,t)}{\partial t^2}$$

Although the there names originate from particular applications, they are frequently used to denote particular mathematical type of equation. As yous ee the mathematical classification here is based on the type of derivatives involved (parabolic/diffusion equation ahs a first order derivative) and the sign of the second order derivatives (Laplace equation ahs all the derivatives of the same sign, whereas the wave equation has one derivative with a different sign.)

Then there are even more domain-specific names, which typically imply certain type of coefficients and/or certain type of inhomogeneous terms. Thus, Poisson equation is an inhomogeneous Laplace equation, whereas Schrödinger equation is a Diffusion equation, with a complex diffusion coefficient, often with a term with zero derivative (the potential term) and potentially also the first derivative term (in magnetic field). Moreover, in some settings Schrödinger equation may contain higher order derivatives, be formulated for multicomponent functions (e.g., in presence of spin), and even for descrete rather than continuous functions (although continuous in time). In other words, it is mathematically and physically very different from the wave equation.

In purely mathematical terms, the second order differential equations are classified into elliptic, parabolic and hyperbolic equations. The most common representatives of these there classes occurring in physics are

• Laplace equation $$\nabla^2 u(x,y,z) = 0$$
• Diffusion equation $$D\nabla^2 u(x,y,z,t) = \frac{\partial u(x,y,z,t)}{\partial t}$$
• Wave equation $$\nabla^2 u(x,y,z,t) = \frac{1}{v^2}\frac{\partial^2 u(x,y,z,t)}{\partial t^2}$$

Although the there names originate from particular applications, they are frequently used to denote particular mathematical type of equation. As yous ee the mathematical classification here is based on the type of derivatives involved (parabolic/diffusion equation ahs a first order derivative) and the sign of the second order derivatives (Laplace equation ahs all the derivatives of the same sign, whereas the wave equation has one derivative with a different sign.)

Then there are even more domain-specific names, which typically imply certain type of coefficients and/or certain type of inhomogeneous terms. Thus, Poisson equation is an inhomogeneous Laplace equation, whereas Schrödinger equation is a Diffusion equation, with a complex diffusion coefficient, often with a term with zero derivative (the potential term) and potentially also the first derivative term (in magnetic field). Moreover, in some settings Schrödinger equation may contain higher order derivatives, be formulated for multicomponent functions (e.g., in presence of spin), and even for descrete rather than continuous functions (although continuous in time). In other words, it is mathematically and physically very different from the wave equation.

In purely mathematical terms, the second order partial differential equations are classified into elliptic, parabolic and hyperbolic equations. The most common representatives of these three classes occurring in physics are

• Laplace equation $$\nabla^2 u(x,y,z) = 0$$
• Diffusion equation $$D\nabla^2 u(x,y,z,t) = \frac{\partial u(x,y,z,t)}{\partial t}$$
• Wave equation $$\nabla^2 u(x,y,z,t) = \frac{1}{v^2}\frac{\partial^2 u(x,y,z,t)}{\partial t^2}$$

Although the there names originate from particular applications, they are frequently used to denote particular mathematical type of equation. As yous ee the mathematical classification here is based on the type of derivatives involved (parabolic/diffusion equation ahs a first order derivative) and the sign of the second order derivatives (Laplace equation ahs all the derivatives of the same sign, whereas the wave equation has one derivative with a different sign.)

Then there are even more domain-specific names, which typically imply certain type of coefficients and/or certain type of inhomogeneous terms. Thus, Poisson equation is an inhomogeneous Laplace equation, whereas Schrödinger equation is a Diffusion equation, with a complex diffusion coefficient, often with a term with zero derivative (the potential term) and potentially also the first derivative term (in magnetic field). Moreover, in some settings Schrödinger equation may contain higher order derivatives, be formulated for multicomponent functions (e.g., in presence of spin), and even for descrete rather than continuous functions (although continuous in time). In other words, it is mathematically and physically very different from the wave equation.