# Quantum harmonic oscillator wave function

• Dean Navels
In summary: You can solve the equation for ##c_n## by expanding in the eigenfunctions of the hamiltonian and solving for ##c_n##.
Dean Navels
How do you find the wave function Φα when given the Hamiltonian, and the equation:

α(x) = αΦα(x)

Where I know the operator

a = 1/21/2((x/(ħ/mω)1/2) + i(p/(mħω)1/2))

And the Hamiltonian,

(p2/2m) + (mω2x2)/2

And α is a complex parameter.

I obviously don't want someone to do this question for me, just a point in the right direction. Thanks in advance!

Do a Google search with terms "harmonic oscillator raising and lowering operators" and you will probably find many web sites where this is discussed in detail.

hilbert2 said:
Do a Google search with terms "harmonic oscillator raising and lowering operators" and you will probably find many web sites where this is discussed in detail.
I've read a lot of them, what I don't understand is that when you apply a lowering or raising operator Φα becomes Φα+1 or Φα-1 yet here it doesn't change.

Suppose the function ##\Phi_\alpha## can be expanded in terms of ##\psi_n##, the eigenstates of the harmonic oscillator hamiltonian:

##\Phi_\alpha (x) = c_0 \psi_0 (x) + c_1 \psi_1 (x) + c_2 \psi_2 (x) + \dots##.

Now form a recursion equation for the numbers ##c_n## from the condition ##a\Phi_\alpha = \alpha\Phi_\alpha##

EDIT: Sorry, I meant "recurrence relation", which is typical English-language term for that.

Last edited:
hilbert2 said:
Suppose the function ##\Phi_\alpha## can be expanded in terms of ##\psi_n##, the eigenstates of the harmonic oscillator hamiltonian:

##\Phi_\alpha (x) = c_0 \psi_0 (x) + c_1 \psi_1 (x) + c_2 \psi_2 (x) + \dots##.

Now form a recursion equation for the numbers ##c_n## from the condition ##a\Phi_\alpha = \alpha\Phi_\alpha##

EDIT: Sorry, I meant "recurrence relation", which is typical English-language term for that.
Is this required to solve it via ladder operators?

^ Yes, you have to use the fact that any quantum state can be expanded in the basis of the eigenfunctions of the hamiltonian, and also the knowledge of how the ladder operators act on those functions.

hilbert2 said:
^ Yes, you have to use the fact that any quantum state can be expanded in the basis of the eigenfunctions of the hamiltonian, and also the knowledge of how the ladder operators act on those functions.
After doing extensive reading on and around it, I know how to find the formula for different energies corresponding to different eigenfunctions, but now I need to put Φα(x) in the form of a normalisation constant C multiplied by an exponential.

The action of a lowering operator ##a## on a QHO eigenstate ##\psi_n## is:

##a\psi_n (x) = \sqrt{n}\psi_{n-1}(x)## .

Now form a general linear combination from the functions ##\psi_n## and act on it with ##a##:

##a\sum_{n=0}^{\infty} c_n \psi_n (x) = \sum_{n=1}^{\infty} \sqrt{n} c_n \psi_{n-1} (x) = \sum_{n=0}^{\infty} \sqrt{n+1} c_{n+1} \psi_{n} (x)## .

Now you should be able to write an equation for ##c_n## in terms of ##c_{n-1}## and ##\alpha##.

hilbert2 said:
The action of a lowering operator ##a## on a QHO eigenstate ##\psi_n## is:

##a\psi_n (x) = \sqrt{n}\psi_{n-1}(x)## .

Now form a general linear combination from the functions ##\psi_n## and act on it with ##a##:

##a\sum_{n=0}^{\infty} c_n \psi_n (x) = \sum_{n=1}^{\infty} \sqrt{n} c_n \psi_{n-1} (x) = \sum_{n=0}^{\infty} \sqrt{n+1} c_{n+1} \psi_{n} (x)## .

Now you should be able to write an equation for ##c_n## in terms of ##c_{n-1}## and ##\alpha##.
I don't understand how to involve alpha. Thanks so much for all your help by the way.

Would I be correct in saying ∑(n-1)^½ Cn-1 Ψn(x)

The eigenfunctions ##\psi_n## are linearly independent, therefore ##\alpha c_n = \sqrt{n+1}c_{n+1}## (why?). So what are the first few terms in the expansion of ##\Phi_\alpha (x)## ?

Dean Navels said:
How do you find the wave function Φα when given the Hamiltonian, and the equation:

α(x) = αΦα(x)

Where I know the operator

a = 1/21/2((x/(ħ/mω)1/2) + i(p/(mħω)1/2))

And the Hamiltonian,

(p2/2m) + (mω2x2)/2

And α is a complex parameter.
See, for example http://quantummechanics.ucsd.edu/ph130a/130_notes/node153.html
The solutions of the Schrödinger equation can be written as a product of a Gaussian with a polynomial. You get recurrence relation among the coeffiients of the polynomials. These polynomials have the name Hermite polynomials.

^ He's not looking for the eigenfunctions of the hamiltonian, the problem is about the eigenfunctions of the lowering operator.

## 1. What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a theoretical system used in quantum mechanics to describe the behavior of a particle in a potential energy well that varies harmonically with position. It is an important model for understanding the behavior of atoms, molecules, and other small particles on a quantum level.

## 2. What is a wave function?

A wave function, also known as a quantum state, is a mathematical description of the quantum state of a system. It is represented by a mathematical function that contains all the information about the possible position, momentum, and energy of a particle at any given time.

## 3. How is the quantum harmonic oscillator wave function derived?

The quantum harmonic oscillator wave function is derived using the Schrödinger equation, which is a fundamental equation in quantum mechanics that describes how the wave function of a quantum system evolves over time. By solving this equation for a harmonic potential, we can derive the specific wave function for a quantum harmonic oscillator.

## 4. What is the significance of the quantum harmonic oscillator wave function?

The quantum harmonic oscillator wave function is significant because it helps us understand the behavior of particles on a quantum level. It allows us to calculate the probability of finding a particle in a particular location or state, and it is also used in many other areas of physics, such as quantum field theory and quantum optics.

## 5. Can the quantum harmonic oscillator wave function be applied to real-world systems?

While the quantum harmonic oscillator is a theoretical model, it has many practical applications in physics and other fields. For example, it is used to describe the behavior of electrons in solid-state physics and the vibrations of molecules in chemistry. While there may be small differences between the theoretical model and real-world systems, the quantum harmonic oscillator wave function provides a good approximation for many physical systems.

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