Schrodinger using a Hermite Polynomial

In summary, the problem asks for confirmation that the 1st excited state wavefunction of a 1D HO given by the Hermite polynomial is a solution of the Schrodinger equation.
  • #1
QuantumMech
16
0
Can some1 help me solve a first energy level Schrodinger ([tex]\psi_{1}[/tex])with a the Hermite polynomial and also show that it equals to [tex]\frac{3}{2}\hbar \omega[/tex]?

I got as far as
[tex]
\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }\frac{\hbar^2}{2 m} \ \pd{\Psi}{x}{2} + V \Psi = \frac{\hbar}{2m} (2N_{1}\frac{x}{\alpha}e^\frac{-1}{2\alpha^2}x^2(\frac{1}{\alpha^2}+\frac{1}{\alpha^4}x^2)
[/tex]

Thanks.
 
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  • #2
Is it for the simple 1D-HO...?That one has Hermite polynomials as eigenfunctions...

If so,how about checking any book on QM (any,all treat 1D-HO) for the famous algebraic method...?

U'll then get the [itex] \psi_{1}(x) [/itex] by applying the creation operator on the ground state...

Daniel.
 
  • #3
The problem says: confirm that the 1st excited state wavefunction of a 1D HO given by the Hermitian equation

[tex] H_{1}(y)= 2y y = \frac{x}{\alpha}, \alpha = (\frac{\hbar^2}{mk})^\frac{1}{4}
[/tex]

is a solution of the Schrodinger equation and that the energy is [tex]\frac{3}{2}\hbar\omega[/tex].


Thanks.
 
Last edited:
  • #4
[tex] \psi_{1}(x)=:\langle x|1\rangle [/tex]


[tex] \hat{H}=\hbar\omega\left(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\hat{1}\right) [/tex]

[tex] \langle x|\hat{H}|1\rangle =\hbar\omega \left(\langle x|\hat{a}^{\dagger}\hat{a}|1\rangle +\frac{1}{2} \langle x|\hat{1}|1 \rangle \right) = \hbar\omega \left\langle x\left|\left(1+\frac{1}{2}\right)\right|1 \right\rangle=\frac{3}{2} \hbar\omega \langle x|1\rangle [/tex]

which means

[tex] \hat{H}\psi_{1}(x)=\left(\frac{3}{2}\hbar\omega\right) \psi_{1}(x) [/tex]

Q.e.d.


NOTE:I made use of

[tex] \left \{\begin{array}{c} \hat{a}|n\rangle =\sqrt{n}|n-1\rangle ,\ \mbox{for} \ n=1 \\ \hat{a}^{\dagger}|n\rangle =\sqrt{n+1}|n+1\rangle ,\ \mbox{for} \ n=0 \end{array} \right [/tex]

which are typical for the operators which form the famous Heisenberg algebra of the 1D-HO.

Daniel.
 
  • #5
Yes... in order to show that a solution satisfies the equation, you don't actually have to solve the equation! Just substitute the proposed solution in and see if the resulting statement is true.
 
  • #6
I don't know bracket notation or what a means, but I put it in the HW I turning in.
 
  • #7
It's the only elegant way to do it,really.

Daniel.
 

Related to Schrodinger using a Hermite Polynomial

What is Schrodinger's equation?

Schrodinger's equation is a mathematical equation that describes the behavior of particles at the quantum level. It is used to predict the probability of finding a particle at a specific location and time.

How does Schrodinger use Hermite polynomials in his equation?

Schrodinger used Hermite polynomials as a basis for his wave function, which describes the probability amplitude of a particle at a given time and location. These polynomials are used to represent the energy levels and spatial distribution of a particle.

Why did Schrodinger choose to use Hermite polynomials?

Schrodinger chose to use Hermite polynomials because they have the unique property of being solutions to the Schrodinger equation. This allows for a more accurate description of a particle's behavior at the quantum level.

How do Hermite polynomials relate to quantum mechanics?

Hermite polynomials are an essential tool in quantum mechanics as they provide a mathematical framework for describing the behavior of particles at the atomic and subatomic level. They are used to calculate the energy levels and probability distributions of particles.

What are the applications of Schrodinger's equation using Hermite polynomials?

Schrodinger's equation using Hermite polynomials has various applications in fields such as quantum mechanics, chemistry, and materials science. It is used to study the behavior of particles in quantum systems and to predict the properties of materials at the atomic level.

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