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Any equation claiming to describe fundamental process has to have time. We know that in nature system changes with time. They are not static. For example, absorption of a photon by a hydrogen atom is a dynamic process and we need to have a theory which can predict this dynamical process in time. Since Schrodinger equation is supposed to be an equation describing nature, time needs to get involved somewhere.

The question then is why did Schrodinger choose this particular equation? There is no rigorous argument to that. If you read Schrodinger's original paper, you will see that Schrodinger used Hamilton Jacobi Equation, $H(q,\frac{\partial S}{\partial q})-\frac{\partial S}{\partial t}=0$. He replaced action $S$ by, $$\psi=e^{KS},$$ where $K$ is some constant with the dimension of action (seems familiar?). He then postulates thenthat integral of the left-hand side of above equation should be extremized. This leads to Schrodinger Equation. The factor of time derivative in Schrodinger equation comes from the term $\frac{\partial S}{\partial t}$ in Hamilton Jacobi Equation.

Above procedure may seem, and is, ad-hoc. But he was able to calculate energy spectrum of Hydrogen atom successfully with it. He also wrote a paper in which he sort of gives motivation for this procedure. He used the geometrical formulation of Hamilton Jacobi Equation and argued that quantum physics differ from classical in the same way wave optics differ from geometric optics.

In short, we require wave function to have time dependence if it has to describe nature. The way this dependence was introduced by Schrodinger is little bit shaky but not entirely bogus. I would suggest you to read his original papers if you want to understand his motivation.

Any equation claiming to describe fundamental process has to have time. We know that in nature system changes with time. They are not static. For example, absorption of a photon by a hydrogen atom is a dynamic process and we need to have a theory which can predict this dynamical process in time. Since Schrodinger equation is supposed to be an equation describing nature, time needs to get involved somewhere.

The question then is why did Schrodinger choose this particular equation? There is no rigorous argument to that. If you read Schrodinger's original paper, you will see that Schrodinger used Hamilton Jacobi Equation, $H(q,\frac{\partial S}{\partial q})-\frac{\partial S}{\partial t}=0$. He replaced action $S$ by, $$\psi=e^{KS},$$ where $K$ is some constant with the dimension of action (seems familiar?). He then postulates then integral of the left-hand side of above equation should be extremized. This leads to Schrodinger Equation. The factor of time derivative in Schrodinger equation comes from the term $\frac{\partial S}{\partial t}$ in Hamilton Jacobi Equation.

Above procedure may seem, and is, ad-hoc. But he was able to calculate energy spectrum of Hydrogen atom successfully with it. He also wrote a paper in which he sort of gives motivation for this procedure. He used the geometrical formulation of Hamilton Jacobi Equation and argued that quantum physics differ from classical in the same way wave optics differ from geometric optics.

In short, we require wave function to have time dependence if it has to describe nature. The way this dependence was introduced by Schrodinger is little bit shaky but not entirely bogus. I would suggest you to read his original papers if you want to understand his motivation.

Any equation claiming to describe fundamental process has to have time. We know that in nature system changes with time. They are not static. For example, absorption of a photon by a hydrogen atom is a dynamic process and we need to have a theory which can predict this dynamical process in time. Since Schrodinger equation is supposed to be an equation describing nature, time needs to get involved somewhere.

The question then is why did Schrodinger choose this particular equation? There is no rigorous argument to that. If you read Schrodinger's original paper, you will see that Schrodinger used Hamilton Jacobi Equation, $H(q,\frac{\partial S}{\partial q})-\frac{\partial S}{\partial t}=0$. He replaced action $S$ by, $$\psi=e^{KS},$$ where $K$ is some constant with the dimension of action (seems familiar?). He then postulates that integral of the left-hand side of above equation should be extremized. This leads to Schrodinger Equation. The factor of time derivative in Schrodinger equation comes from the term $\frac{\partial S}{\partial t}$ in Hamilton Jacobi Equation.

Above procedure may seem, and is, ad-hoc. But he was able to calculate energy spectrum of Hydrogen atom successfully with it. He also wrote a paper in which he sort of gives motivation for this procedure. He used the geometrical formulation of Hamilton Jacobi Equation and argued that quantum physics differ from classical in the same way wave optics differ from geometric optics.

In short, we require wave function to have time dependence if it has to describe nature. The way this dependence was introduced by Schrodinger is little bit shaky but not entirely bogus. I would suggest you to read his original papers if you want to understand his motivation.

Source Link
utagr
utagr
  • 91
  • 5

Any equation claiming to describe fundamental process has to have time. We know that in nature system changes with time. They are not static. For example, absorption of a photon by a hydrogen atom is a dynamic process and we need to have a theory which can predict this dynamical process in time. Since Schrodinger equation is supposed to be an equation describing nature, time needs to get involved somewhere.

The question then is why did Schrodinger choose this particular equation? There is no rigorous argument to that. If you read Schrodinger's original paper, you will see that Schrodinger used Hamilton Jacobi Equation, $H(q,\frac{\partial S}{\partial q})-\frac{\partial S}{\partial t}=0$. He replaced action $S$ by, $$\psi=e^{KS},$$ where $K$ is some constant with the dimension of action (seems familiar?). He then postulates then integral of the left-hand side of above equation should be extremized. This leads to Schrodinger Equation. The factor of time derivative in Schrodinger equation comes from the term $\frac{\partial S}{\partial t}$ in Hamilton Jacobi Equation.

Above procedure may seem, and is, ad-hoc. But he was able to calculate energy spectrum of Hydrogen atom successfully with it. He also wrote a paper in which he sort of gives motivation for this procedure. He used the geometrical formulation of Hamilton Jacobi Equation and argued that quantum physics differ from classical in the same way wave optics differ from geometric optics.

In short, we require wave function to have time dependence if it has to describe nature. The way this dependence was introduced by Schrodinger is little bit shaky but not entirely bogus. I would suggest you to read his original papers if you want to understand his motivation.