# Understanding the Terms in the 1-D Schrodinger Equation

• zcapa14
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.
zcapa14
what is the physical significance of each of the terms in the 1-D time indipendant schrodinger equation?

Why are u asking us...?What are your ideas...?What u've read,i presume it's not a curiosity,but some sort of homework.

Daniel.

The question is from last years exam that i am doing for revision. however i have no answers, my notes are a little sketchy when they get round to schrodinger equation and QMT...

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.

I see.There are two ways of looking at it.Traditional way,in which the equation is postulated and everything is deduces from there,or the symmetry way,i'd like to call it J.J.Sakurai way.

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.

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

I think he said time-independent, as opposed to time-dependent. Furthermore, the two approaches you have mentioned look exactly the same to me. One of them says that the rate of change of the state vector is given by the Hamiltonian acting on the state vector, the other says the Hamiltonian (is self-adjoint, but that's obvious anyway) generates infinitesimal translations in time, which is the rate of change.

$$\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi = E\Psi$$

If you divide out $$\Psi$$ 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 $$\frac{p^2}{2m}$$ is replaced by $$\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$$ in QM.

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

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

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

$$\frac{-\hbar^2\nabla^2}{2m}\Psi$$

as the momentum operator is:

$$\hat p=-i\hbar\nabla$$

Last edited:
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.

## What is the 1-D Schrodinger Equation?

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.

## What are the terms in the 1-D Schrodinger Equation?

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.

## What is the wave function in the 1-D Schrodinger Equation?

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.

## How is the 1-D Schrodinger Equation used in quantum mechanics?

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.

## What are the assumptions made in the 1-D Schrodinger Equation?

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.

Replies
8
Views
2K
Replies
9
Views
2K
Replies
2
Views
871
Replies
5
Views
1K
Replies
17
Views
2K
Replies
1
Views
1K
Replies
1
Views
979
Replies
143
Views
8K
Replies
24
Views
2K
Replies
5
Views
1K