The Heisenberg uncertainty principle can be expressed in the form $$\Delta E \cdot \Delta t \geq \frac{h}{4 \pi}$$ where \(E\) represents energy and \(t\) represents time. Show that the units for this form are the same as the units for the form used in this chapter: $$\Delta x \cdot \Delta(m v) \geq \frac{h}{4 \pi}$$

First, we should determine the units for each variable in SI units: - \(E\): Energy, measured in Joules (J) - \(t\): Time, measured in seconds (s) - \(x\): Position, measured in meters (m) - \(m\): Mass, measured in kilograms (kg) - \(v\): Velocity, measured in meters per second (m/s) - \(h\): Planck's constant, measured in Joule-seconds (J·s)

Now that we have determined the units for each variable, let's analyze the units for each term in the first form of Heisenberg Uncertainty Principle: $$\Delta E \cdot \Delta t \geq \frac{h}{4 \pi}$$ Substitute the SI units for the variables: $$\Delta(\text{J}) \cdot \Delta(\text{s}) \geq \frac{\text{J} \cdot \text{s}}{4 \pi}$$ Simplify: $$\text{J} \cdot \text{s} \geq \frac{\text{J} \cdot \text{s}}{4 \pi}$$

Now let's analyze the units for each term in the second form of Heisenberg Uncertainty Principle: $$\Delta x \cdot \Delta(mv) \geq \frac{h}{4 \pi}$$ Substitute the SI units for the variables: $$\Delta(\text{m}) \cdot \Delta(\text{kg} \cdot \frac{\text{m}}{\text{s}}) \geq \frac{\text{J} \cdot \text{s}}{4 \pi}$$ Simplify: $$\text{m} \cdot (\text{kg} \cdot \text{m}/\text{s}) \geq \frac{\text{J} \cdot \text{s}}{4 \pi}$$ Now, we need to convert the left-hand side of this inequality to Joule-seconds. Recall that the relationship between Joules, kilograms, and meters is given by the formula \(E = \frac{1}{2}mv^2\). So, we can substitute \(\text{J} = \text{kg} \cdot (\text{m}/\text{s})^2\): $$\text{m} \cdot (\text{kg} \cdot \text{m}/\text{s}) = \text{J} \cdot \text{s}$$

From steps 2 and 3, we have the following inequalities: Inequality 1: $$\text{J} \cdot \text{s} \geq \frac{\text{J} \cdot \text{s}}{4 \pi}$$ Inequality 2: $$\text{J} \cdot \text{s} \geq \frac{\text{J} \cdot \text{s}}{4 \pi}$$ It can be seen that the units on both sides of the inequalities in step 2 and 3 are the same, which means that both forms of the Heisenberg Uncertainty Principle have the same units and are consistent with each other.