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Consider a quantum free particle characterized by the complex plane wave . Show that the free particle satisfies the Schrodinger equation for some potential function .

Quantum particle-waves.jpg


The Hamiltonian of a system is where is the kinetic energy and is the potential energy. Since the quantity is the total energy, let us rewrite the Hamiltonian as .

Now, we take the derivatives:

Since and , where is the wavenumber and is the wavelength, we have .


Next, we multiply to the Hamiltonian:

Notice that the above equation can be expressed as

Since the energy of matter waves is given by , we can show that

Combining all the right parts, we assemble the Schrödinger equation: