Heisenberg's & Schrodinger's Picture

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In summary, the Heisenberg picture and Schrödinger picture both produce the same results, but treat space and time differently. The Heisenberg picture is more congruous with relativity and quantum field theory, as it allows for particle creation and annihilation. The two pictures become compatible in the non-relativistic formulation, with the Schrödinger picture corresponding to a lab-fixed reference frame and the Heisenberg picture corresponding to a system-fixed reference frame.
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Eynstone
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Is the Heisenberg's picture really equivalent to Schrodinger's ?
It may seem so at a first glance & both produce the same results.But space & time are treated equally in Heisenberg's formulation : the operators are time dependent.In Schrodinger's formulation, states are time-dependent while the hamiltonian may be time-independent. This makes the former congruous & adaptable to relativity.
Also, the initial state vector remains fixed & operators move in Heisenberg's formulation. The initial state is not of so great a consequence as in Schrodinger's picture. Hence, it is more congruous with quantum field theory .(When particle creation & annihilation is involved, we can't work with a state vector in a Hilbert space).
I wonder how the two become totally compatible in the non-relativistic formulation.
 
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It is easy to show mathematically that the two pictures give equal results.

A simple ways to visualize it is to consider that the Schrödinger picture corresponds to a lab-fixed reference frame, in which the state of the quantum system will evolve, while the Heisenberg picture corresponds to a system-fixed reference frame, where the state remains unchanged but the world around (including measuring apparatuses) move as a function of time.
 

1. What is Heisenberg's Picture?

Heisenberg's Picture is a formulation of quantum mechanics that was developed by Werner Heisenberg in 1925. It is one of the two basic formulations of quantum mechanics, the other being Schrodinger's Picture. In this formulation, the state of a quantum system is described by quantum operators that evolve with time, while the observables, such as position and momentum, are fixed.

2. What is Schrodinger's Picture?

Schrodinger's Picture is a formulation of quantum mechanics that was developed by Erwin Schrodinger in 1926. It is one of the two basic formulations of quantum mechanics, the other being Heisenberg's Picture. In this formulation, the state of a quantum system is described by a wave function that evolves with time, while the observables, such as position and momentum, are represented by operators.

3. What is the difference between Heisenberg's and Schrodinger's Picture?

The main difference between Heisenberg's and Schrodinger's Picture lies in their approach to representing quantum systems. In Heisenberg's Picture, the state of a system is represented by operators while the observables are fixed. In Schrodinger's Picture, the state of a system is represented by a wave function while the observables are represented by operators. Additionally, the time evolution in Heisenberg's Picture is given by the operators, while in Schrodinger's Picture it is given by the wave function.

4. Which picture is more commonly used in quantum mechanics?

Both Heisenberg's and Schrodinger's Picture are used in quantum mechanics, but Schrodinger's Picture is more commonly used in introductory courses and calculations. However, Heisenberg's Picture is often used in advanced calculations and in the study of quantum field theory.

5. How are Heisenberg's and Schrodinger's Picture related?

Heisenberg's and Schrodinger's Picture are two different formulations of quantum mechanics, but they are mathematically equivalent. This means that one can be derived from the other, and they both provide the same predictions for the behavior of quantum systems. They are just different ways of representing and understanding quantum mechanics.

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