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zcapa14

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- Thread starter zcapa14
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In summary, the 1-D time independent Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of a system with respect to time. It includes terms for kinetic energy, potential energy, and total energy, and can be derived through two different approaches. V represents the potential energy of the system, which is related to any conservative forces acting on it.

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zcapa14

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Daniel.

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zcapa14

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i am aware of the definition of each of the terms, but the question wants more than this, namely the 'physical significance' of each of these terms. That is what i am puzzled at.

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So the trick is simple.The axiomatical approach asserts that the speed of variation in time of the state vector is proportional to the hamiltonian applied to that state vector.The other way,is to derive this equation by stating that the Hamiltonian of the system is the self-adjoint generator of the abelian group of time translations...

So,for further reference for this interesting symmetry-based approach,i invite you to read the second chapter (i think the 2-nd or the 3-rd section) from [1].

Daniel.

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[1]J.J.Sakurai,"Modern Quantum Mechanics",Addison-Wesley,any of the 2 editions.

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masudr

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werty

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[tex]\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi = E\Psi[/tex]

If you divide out [tex]\Psi[/tex] then the first term could be interpreted as the kinetic energy, the second potential energy and the right side as the total energy. So basically its a statement about energy conservation.

Remember that [tex]\frac{p^2}{2m}[/tex] is replaced by [tex]\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}[/tex] in QM.

Offcourse its not legal to divide out the [tex]\Psi[/tex], so the first term is some kinetic energy times a probability density function, and so on.

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smoslemi

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Could you explan what is V (the potential energy)?

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James Jackson

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Well, that's in 1D cartesian. Being more rigorous, the first term becomes

[tex]\frac{-\hbar^2\nabla^2}{2m}\Psi[/tex]

as the momentum operator is:

[tex]\hat p=-i\hbar\nabla[/tex]

[tex]\frac{-\hbar^2\nabla^2}{2m}\Psi[/tex]

as the momentum operator is:

[tex]\hat p=-i\hbar\nabla[/tex]

Last edited:

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James R

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Could you explan what is V (the potential energy)?

V is the potential energy related to some kind of conservative force acting on the system to which you are applying the Schrodinger equation.

For example, if you're solving the equation for the hydrogen atom, V is the electrostatic potential energy of the proton and the electron.

The 1-D Schrodinger Equation is a mathematical equation that describes the behavior of quantum particles in one dimension. It is a fundamental equation in quantum mechanics and is used to calculate the probability of finding a particle at a given position and time.

The 1-D Schrodinger Equation consists of three main terms: the kinetic energy term, the potential energy term, and the time-dependent term. These terms represent the energy, the potential energy of the particle in a given environment, and the change in the wave function over time, respectively.

The wave function in the 1-D Schrodinger Equation is a mathematical function that describes the quantum state of a particle. It represents the probability amplitude of finding the particle at a given position and time.

The 1-D Schrodinger Equation is used in quantum mechanics to predict the behavior of quantum particles in one dimension. It is used to calculate the probability of finding a particle at a given position and time, and it is also used to solve for the energy levels of quantum systems.

The 1-D Schrodinger Equation relies on several assumptions, including the particle being in one dimension, the potential energy being time-independent, and the particle not interacting with other particles. These assumptions allow for the equation to accurately describe the behavior of quantum particles in a simplified manner.

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