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Write down the Hamiltonian operator Hˆ for a single particle of mass m in 1 dimension...

Write down the Hamiltonian operator Hˆ for a single particle of mass m in 1 dimension in a potential U(x). Identify the term that represents kinetic energy. Identify the term that represents potential energy.

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Answer #1

Let's start from the definition, Hamiltonian is the sum of Kinetic energy and Potential energy.

So we have,

Since the particle is of mass "m" and in one dimension we can write Kinetic energy as-

And Potential energy can be written as-

We know in Quantum mechanics, we write momentum operator in one dimension (so it only depends on x) as-

So, our final form of Hamiltonian operator becomes-

Here first term represents Kinetic energy and the second term represents the potential energy.

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