Chapter 7, Problem 7D.7P

(a)

Interpretation Introduction

Interpretation:

The expression for $\Delta x$ and $\Delta p_x$ for a particle of mass $m$ in a one dimensional box of length $L$ has to be calculated.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by $\psi$ in the quantum mechanics.  The normalized wavefunction of a particle of mass $m$ in a one dimensional box of length $L$ for a general quantum state $n$ is shown below.

    $\psi ={\left(\frac{2}{L}\right)}^{1/2}\sin \left(n\pi x/L\right)$

(b)

Interpretation Introduction

Interpretation:

The expression for the product of $\Delta x\Delta p_x$ for a particle of mass $m$ in a one dimensional box of length $L$ has to be calculated.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by $\psi$ in the quantum mechanics.  The normalized wavefunction of a particle of mass $m$ in a one dimensional box of length $L$ for a general quantum state $n$ is shown below.

    $\psi ={\left(\frac{2}{L}\right)}^{1/2}\sin \left(n\pi x/L\right)$

(c)

Interpretation Introduction

Interpretation:

The validation of Heisenberg’s uncertainty principle for $n=1,n=2$ and also for $n\ge 1$ has to be shown.

Concept introduction:

Heisenberg’s uncertainty principle states that for a microscopic particle, there is always some uncertainty present in the measurement of the uncertainty in momentum and the uncertainty in the position of a particle simultaneously.  Heisenberg’s uncertainty principle can be proved indirectly by showing the result of the product of $\Delta x\Delta p_x$ a positive quantity.