Explanation The Schrödinger’s equation is given as, − ℏ 2 2 m d 2 ψ ( x ) d x 2 + 1 2 k f x 2 ψ ( x ) = E ψ ( x ) (1) The given equation is, ψ 1 ( x ) = N 1 ( 2 α ) x e − x 2 / 2 α 2 (2) The second derivative of the excited state wavefunction is calculated by differentiating it twice as, d 2 d x 2 ψ 1 = d 2 d x 2 ( N 1 ( 2 α ) x e − x 2 / 2 α 2 ) The first derivative of the wavefunction is calculated as, d d x ( N 1 ( 2 α ) x e − x 2 / 2 α 2 ) = N 1 ( 2 α ) ( d d x ( x e − x 2 / 2 α 2 ) ) = N 1 ( 2 α ) ( e − x 2 / 2 α 2 d x d x + x d d x ( e − x 2 / 2 α 2 ) ) = N 1 ( 2 α ) ( e − x 2 / 2 α 2 + x e − x 2 / 2 α 2 ( − 2 x 2 α 2 ) ) Differentiating the first derivative obtained above a second time, N 1 ( 2 α ) d d x ( e − x 2 / 2 α 2 − x 2 α 2 e − x 2 / 2 α 2 ) = N 1 ( 2 α ) ( d d x ( e − x 2 / 2 α 2 ) − d d x ( x 2 α 2 e − x 2 / 2 α 2 ) ) = N 1 ( 2 α ) ( e − x 2 / 2 α 2 ( − 2 x 2 α 2 ) − e − x 2 / 2 α 2 α 2 d d x x 2 − x 2 α 2 d d x e − x 2 / 2 α 2 ) = N 1 ( 2 α ) ( − x α 2 e − x 2 / 2 α 2 − 2 x α 2 e − x 2 / 2 α 2 − x 2 α 2 e − x 2 / 2 α 2 ( − 2 x 2 α 2 ) ) = N 1 ( 2 α ) ( 1 α 2 ) ( − 3 x e − x 2 / 2 α 2 + x 3 α 2 e − x 2 / 2 α 2 ) Therefore, d 2 d x 2 ψ 1 = N 1 ( 2 α ) ( 1 α 2 ) ( − 3 x e − x 2 / 2 α 2 ) + N 1 ( 2 α ) ( 1 α 2 ) x 3 α 2 e − x 2 / 2 α 2 (3) Substitute ψ 1 from equation (2) in equation (3)

11th Edition

ISBN: 9780191092183

Author: ATKINS

Publisher: OXFORD UNIV.PRESS ACAD.UK (CC)

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Techniques and applications of quantum theory

Quantum theory, often known as quantum physics, is the foundation of modern physics and encompasses the nature and behavior of matter and energy at the atomic and molecular levels. As the exact nature and behavior of atomic and subatomic particles are im…

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Chapter 7, Problem 7E.1ST

Interpretation Introduction

Interpretation:

Whether the given wavefunction is a solution of Schrodinger’s equation has to be stated and the energy has to be evaluated.

Concept introduction:

In a harmonic oscillator, a particle of mass m experiences a restoring force which is proportional to the displacement of the particle from equilibrium position. The Schrödinger’s equation is a form of an Eigen function in which the value of the function is equal to the energy and is the Eigen value.

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Chapter 7 Solutions

Atkins' Physical Chemistry

Ch. 7 - Prob. 7A.1ST Ch. 7 - Prob. 7A.2ST Ch. 7 - Prob. 7A.3ST Ch. 7 - Prob. 7B.1ST Ch. 7 - Prob. 7B.2ST Ch. 7 - Prob. 7B.3ST Ch. 7 - Prob. 7C.1ST Ch. 7 - Prob. 7C.2ST Ch. 7 - Prob. 7C.3ST Ch. 7 - Prob. 7C.4ST

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