Physics Friday 55

Classically, a one-dimensional harmonic oscillator is a system with a mass under a restoring force proportional to displacement from the equilibrium position: F=-kx. The energy is , and the equation of motion has solution , where.

Analogously, a one-dimensional quantum harmonic oscillator is a particle with Hamiltonian , where here p is the quantum momentum operator, and x the position operator. In the position basis, this is then
A cursory examination of the expectation for energy

shows that we can expect our energies to be non-negative.

Most quantum mechanics books and courses I have encountered address the energy eigenstates by means of the raising and lowering operators. Instead, here we will use an analytical method of finding the eigenvalues to the time-independent Schrödinger equation
Let us replace x and E with corresponding dimensionless variables: we define the dimensionless variables , . Then ,
the Schrödinger equation becomes

To find a solution to this differential equation, we consider asymptotic behavior. Namely, when the energy is small, the ξ2 dominates over ε, and our equation approaches , which hints at a solution that behaves like a Gaussian. Adopting, then, a test form
, and plugging this into our differential equation , we get
or, eliminating the common Gaussian factor,

Applying the power series method, we plug in to find

Which gives recursion relation .

Now, we return to the physics of the problem, namely that the wavefunctions have to be normalized. This means that ψ→0 as x→±∞. Now, considering that
, and comparing to the series expansion of the gaussian, we see our series diverges too fast for ψ to be normalized, unless the series terminates. This requires for some n, which gives , and as E≥0, we see that the ground state has energy ; this is known as the ground state energy, or zero-point energy.

Solving for the wavefunctions themselves requires more mathematics, and gives solutions involving the Hermite polynomials.


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2 Responses to “Physics Friday 55”

  1. Monday Math 56 « Twisted One 151’s Weblog Says:

    […] includes a number of notable polynomial sequences (including the Hermite polynomials mentioned in a recent Physics Friday post). Note that the sequence Pn(x)=xn is a trivial example. Now, we can obtain from the relation , or […]

  2. Physics Friday 58 « Twisted One 151’s Weblog Says:

    […] Friday 58 By twistedone151 In a previous Friday post, I demonstrated one method of determining the energy eigenvalues for the one-dimensional quantum […]

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