Tru Physics

Time-Dependent Schrödinger Equation

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Introduction

The Time-dependent Schrödinger equation (TDSE) is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes with time. It is named after Erwin Schrödinger, who formulated the equation in 1926.

The Time-dependent Schrödinger Equation

The TDSE is given by:

i \hbar \dfrac{\partial}{\partial t} \Psi(\vec{r},t) = \hat{H} \Psi(\vec{r},t)

Here, \Psi(\vec{r},t) is the wavefunction of the system at position \vec{r} and time t, \hat{H} is the Hamiltonian operator representing the total energy of the system (kinetic plus potential), i is the imaginary unit, and \hbar is the reduced Planck’s constant.

Physical Interpretation of the Time-Dependent Schrödinger Equation

The TDSE provides the temporal evolution of a quantum system’s wavefunction. The square of the modulus of the wavefunction, |\Psi(\vec{r},t)|^2, gives the probability density for the system to be found at position \vec{r} at time t.

Hamiltonian Operator

The Hamiltonian operator in the TDSE is given by:

\hat{H} = -\dfrac{\hbar^2}{2m} \nabla^2 + V(\vec{r},t)

where m is the mass of the particle, V(\vec{r},t) is the potential energy, and \nabla^2 is the Laplacian operator which represents the second spatial derivative.

Time-Independent Schrödinger Equation

For systems with a time-independent Hamiltonian, the Schrödinger equation simplifies to the time-independent Schrödinger equation:

\hat{H} \psi(\vec{r}) = E \psi(\vec{r})

Here, \psi(\vec{r}) is the time-independent wavefunction, and E is the total energy of the system, which is a constant.

Conclusion

The TDSE is a central equation in quantum mechanics, determining the dynamics of quantum systems. It encapsulates key quantum phenomena such as superposition and tunneling. Despite its apparent simplicity, the TDSE often leads to complex behavior that is at the heart of many modern technologies, from semiconductors to quantum computers.

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