Brief Derivation of Schrödinger Equation: Difference between revisions
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Now by applying the appropriate initial conditions for our particle, we obtain a solution for the trajectory of the particle. As we will see, the above relation is only an approximation to actual physical reality. As we attempt to describe increasingly smaller objects we enter the quantum mechanical regime, where we cannot neglect the particles' wave properties. Allowing <math>\displaystyle{p \rightarrow \frac{\hbar}{i}\frac{\partial}{\partial x}}</math> and <math>\displaystyle{H \rightarrow i\hbar \frac{\partial}{\partial t}}</math>, we can use the Hamiltonian for a classical particle above to find an equation that describes this wave nature. Thus, we find that the complex amplitude satisfies the [[Schrödinger equation]] for a scalar potential <math>V(x,t)\!</math> in one dimension: | Now by applying the appropriate initial conditions for our particle, we obtain a solution for the trajectory of the particle. As we will see, the above relation is only an approximation to actual physical reality. As we attempt to describe increasingly smaller objects we enter the quantum mechanical regime, where we cannot neglect the particles' wave properties. Allowing <math>\displaystyle{p \rightarrow \frac{\hbar}{i}\frac{\partial}{\partial x}}</math> and <math>\displaystyle{H \rightarrow i\hbar \frac{\partial}{\partial t}}</math>, we can use the Hamiltonian for a classical particle above to find an equation that describes this wave nature. Thus, we find that the complex amplitude satisfies the [[Schrödinger equation]] for a scalar potential <math>V(x,t)\!</math> in one dimension: | ||
:<math> i\hbar\frac{\partial}{\partial t}\ | :<math> i\hbar\frac{\partial}{\partial t}\Psi(x,t)=\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x,t)\right]\Psi(x,t) </math> | ||
A similar equation may be derived in multiple dimensions: | A similar equation may be derived in multiple dimensions: | ||
:<math> i\hbar\frac{\partial}{\partial t}\ | :<math> i\hbar\frac{\partial}{\partial t}\Psi(\textbf{r},t)=\left[-\frac{\hbar^2}{2m}\nabla^2+V(\textbf{r},t)\right]\Psi(\textbf{r},t)</math> | ||
Given a solution which satisfies the above Schrödinger equation, Quantum Mechanics provides a mathematical description of the laws obeyed by the probability amplitudes associated with quantum motion. | Given a solution which satisfies the above Schrödinger equation, Quantum Mechanics provides a mathematical description of the laws obeyed by the probability amplitudes associated with quantum motion. | ||
We can also generalize the Schrödinger equation to a system which contains <math> N \!</math> particles. We assume that the wave function is <math> \ | We can also generalize the Schrödinger equation to a system which contains <math> N \!</math> particles. We assume that the wave function is <math> \Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t) </math> and the [[Commutation relations and simultaneous eigenvalues#Hamiltonian|Hamiltonian operator]] of the system can be expressed as: | ||
:<math> H= \sum_{k=1}^N \frac{\textbf{p}^2_k}{2m_k}+V(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N) </math> | :<math> H= \sum_{k=1}^N \frac{\textbf{p}^2_k}{2m_k}+V(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N) </math> | ||
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So the Schrödinger equation for a many-particle system is: | So the Schrödinger equation for a many-particle system is: | ||
:<math> i\hbar\frac{\partial}{\partial t}\ | :<math> i\hbar\frac{\partial}{\partial t}\Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t)=\left[\sum_{k=1}^N \frac{\textbf{p}^2_k}{2m_k}+V(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N)\right]\Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t) </math> |
Revision as of 16:38, 8 April 2013
Imagine a particle constrained to move along the -axis, subject to a potential energy . Classically, we could model this system by writing down its Hamiltonian , given by
We then employ Hamilton's equations of motion,
where a dot denotes a time derivative, to determine the motion of the particle. Applying these equations to the above Hamiltonian, we can recover Newton's second law,
Now by applying the appropriate initial conditions for our particle, we obtain a solution for the trajectory of the particle. As we will see, the above relation is only an approximation to actual physical reality. As we attempt to describe increasingly smaller objects we enter the quantum mechanical regime, where we cannot neglect the particles' wave properties. Allowing and , we can use the Hamiltonian for a classical particle above to find an equation that describes this wave nature. Thus, we find that the complex amplitude satisfies the Schrödinger equation for a scalar potential in one dimension:
A similar equation may be derived in multiple dimensions:
Given a solution which satisfies the above Schrödinger equation, Quantum Mechanics provides a mathematical description of the laws obeyed by the probability amplitudes associated with quantum motion.
We can also generalize the Schrödinger equation to a system which contains particles. We assume that the wave function is and the Hamiltonian operator of the system can be expressed as:
So the Schrödinger equation for a many-particle system is: