# Time invariance of Schrodinger equation

• Avijeet
In summary, the Schrodinger equation is linear in time and is therefore time invariant. This may seem surprising since other microscopic laws such as Maxwell's equations and Newton's equations are also time invariant. However, the Schrodinger equation does not respect relativistic invariance, unlike the Dirac equation which uses first order derivatives. This is because the time reversal operator for the Schrodinger equation is antiunitary, meaning that the wave function must be complex-conjugated in addition to changing the sign of time.
Avijeet
The Schrodinger equation is linear in time. I was wondering if that means that is not invariant under time reversal. That would be a surprise because all other microscopic laws (Maxwell's equations, Newton's equations) are time invariant.
Can you please clear this doubt?

Avijeet said:
The Schrodinger equation is linear in time. I was wondering if that means that is not invariant under time reversal. That would be a surprise because all other microscopic laws (Maxwell's equations, Newton's equations) are time invariant.
And it does not respect any more the relativistic invariance.
Only the Dirac equation does. It only uses first order derivatives.

Actually it is time invariant, since the time reversal operator is not unitary but antiunitary, so you have to complex-conjugate the wave function besides changing the sign of the time.
Of course it doesn't respect relativistic invariance.

## 1. What is the concept of time invariance in the Schrodinger equation?

The time invariance of the Schrodinger equation refers to the property that the equation remains the same regardless of the time at which it is solved. In other words, the time at which the equation is solved does not affect the results or solutions.

## 2. Why is time invariance important in quantum mechanics?

Time invariance is important in quantum mechanics because it allows us to make predictions and calculations about a system at any given time. This property also ensures that the principles of quantum mechanics remain consistent and accurate.

## 3. How does the Schrodinger equation demonstrate time invariance?

The Schrodinger equation is a mathematical representation of the time evolution of a quantum system. It is time invariant because it remains unchanged when the time variable is replaced with a different value. This means that the equation gives the same results regardless of the time at which it is solved.

## 4. Are there any exceptions to the time invariance of the Schrodinger equation?

There are certain cases where the Schrodinger equation may not exhibit time invariance, such as when external forces or perturbations are acting on the system. In these situations, the equation may need to be modified to account for these factors.

## 5. How does time invariance relate to the conservation of energy?

The time invariance of the Schrodinger equation is closely related to the conservation of energy in quantum systems. Since the equation remains unchanged over time, it ensures that energy is conserved and that the system's total energy remains constant throughout its evolution.

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