Chapter 4: Q37E (page 136)

A free particle is represented by the plane wave function $ψ (x, t) = A e x p [i (1.58 \times 10^{12} x - 7.91 \times 10^{16} t)]$ where SIunits are understood. What are the particle’s momentum, Kinetic energy, and mass? (Note: In nonrelativistic quantum mechanics, we ignore mass/internal energy, so the frequency is related to kinetic energy alone.)

Short Answer

Expert verified

Particle’s Momentum of Plane Wave Function 1.66x10^-22 kg^.m/s

Particle’s Momentum of Kinetic Energy 8.30x10^-18 J

Particle’s Momentum of Mass 1.66x10^-27s

Step by step solution

Given information

The given wavefunction is $ψ (x, t) = A e x p [i (1.58 \times 10^{12} x - 7.91 \times 10^{16} t)]$

Concept Introduction

The equation of an electromagnetic wave can be expressed as,

$ψ (x, t) = A e^{i (k x - ω t)}$ …………(1)

Calculations for momentum

Compare the given wavefunction with equation (1), and we get

$k = 1.58 \times 10^{12} m^{- 1} ω = 7.91 \times 10^{16} s^{- 1}$

We use the following formula to find the momentum, p :

$\begin{array}{rcl} p & = & h k \\ = & (1.05 \times 10^{- 34} J \cdot s) \times (1.58 \times 10^{12} m^{- 1}) \\ = & 1.66 \times 10^{- 22} k g \cdot m / s \end{array}$

Calculate the Kinetic energy.

To calculate the kinetic energy, , we use:

$\begin{array}{rcl} K E & = & h ω \\ = & (1.05 \times 10^{- 34} J \cdot s) \times (7.91 \times 10^{16} s^{- 1}) \\ = & 8.30 \times 10^{- 18} J \end{array}$

Calculate the mass.

For calculation of the Mass , we use:

$m = \frac{p^{2}}{2 \times K E}$

Substitute the obtained values in the above expression and we get,

$\begin{array}{rcl} m & = & \frac{{(1.66 \times 10^{- 22} k g \cdot m / s)}^{2}}{2 \times (8.3 \times 10^{- 18} J)} \\ = & 1.66 \times 10^{- 27} s \end{array}$

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Short Answer

Step by step solution

Given information

Concept Introduction

Calculations for momentum

Calculate the Kinetic energy.

Calculate the mass.

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