The **harmonic Oscillator wave function** is given energy state like –**Ground state energy wavefunction** (when n=0), **First excited state energy correspond wave function** (n=1) is easily solved it using quantum mechanics.

The chapter on quantum chemistry is very important for chemistry students such as research fellows, entrance examination aspirants, and any degree course students. We have to solve the **Ground state energy wavefunction 𝜓_{0} (𝑥) **and

**First excite state energy wave function 𝜓**. We can see that: how to find out the Harmonic Oscillator wavefunction? This work is easily solved by the Hermite Polynomials equation.

_{1}(𝑥)### The basic Principle of the Harmonic-Oscillator wave function involves Hermite polynomials.

we have to consider a Harmonic-Oscillator energy state blowing wave function involving Hermite polynomials. A particle oscillated in a harmonic way to solve the wave function Using the Hermite Polynomials factor. In the Harmonic Oscillator, the wave function corresponding to the 𝐸_{𝑛} is non-degenerate and is given…

𝜓_{𝑛} (𝑥) = 𝑁_{𝑛} 𝐻_{𝑛} (𝛼^(1∕2) 𝑥)𝑒^(−𝛼𝑥^{2}/2) …..(i)

Where 𝛼=〖(𝑘𝜇/ℏ^2 )〗^(1/2)

**Hermite Polynomials = 𝐻 _{𝑛} (𝛼^{1∕2} 𝑥)**

The Normalization constant 𝑁_{𝑛} is N_{n}=1/〖(2^n n!)〗^(1/2) 〖(α/π)〗^(1/4)

### First few Hermite Polynomials:

𝐻_{0} (𝜉)=1

𝐻_{1} (𝜉)=2(𝜉)

and 𝐻_{2} (𝜉)=4𝜉^{2}−2

### Question and Answer discussion:

Now we have to discuss the Questions with solutions on Harmonic Oscillator wavefunction regards Quantum chemistry. Easily solve these questions if you have knowledge or a clear concept of Harmonic Oscillator wavefunction. The question asked in objective type: but the concept is the same as MCQ. You can try to **solve the Harmonic Oscillator wavefunction involving Hermite polynomials questions**.

Use the practice questions answer on quantum chemistry for any entrance or national level examination. The section of Quantum chemistry part-3. We have to earlier discuss the **Rotational Spectroscopy approach of quantum chemistry** and **Commutator Operator**. Follow the Question…

#### Q.3. Show that 𝝍_{𝟎} (𝒙) 𝒂𝒏𝒅 𝝍_{𝟏} (𝒙) normalized. Where Ground state energy wave function and first excited state energy wave function of Harmonic Oscillator 𝝍_{𝟎} (𝒙) 𝒂𝒏𝒅 𝝍_{𝟏} (𝒙) Respectively.

Answer:

We have to know that the Harmonic oscillator ground state wavefunction is 𝝍_{𝟎} (𝒙). And First excited state energy wavefunction is 𝝍_{1} (𝒙). Two equations are get from eq. no. (i). When n =0 ground state and n =1 excited state wave functions are respectively.

The Harmonic oscillator ground state wavefunction is 𝝍_{𝟎} (𝒙) and First excited state energy wavefunction is 𝝍_{1} (𝒙). |

Now, we have to show the Normalized wave function 𝝍_{𝟎} (𝒙) 𝒂𝒏𝒅 𝝍_{𝟏} (𝒙).

For the normalized function ∫ Ψ^{∗}(𝑥) Ψ(𝑥) 𝑑𝑥 =1

So, finally, we have to prove that 𝝍_{𝟎} (𝒙) 𝒂𝒏𝒅 𝝍_{𝟏} (𝒙) normalized. Where Ground state energy wave function and first excited state energy wave function of Harmonic Oscillator 𝝍_{𝟎} (𝒙) 𝒂𝒏𝒅 𝝍_{𝟏} (𝒙) Respectively.

### Reference:

Chemclip suggests a few readable and conceptual, approaches to clearing books or referral sources. That is helpful for beginners corresponds to advance level students. The books are part of the **Physical Chemistry Books’ suggestion**.

- Quantum chemistry book by Donald A McQuarrie (Viva student edition)
- Quantum Mechanics Concepts and Applications second edition by Nouredine Zettili.

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