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, let's recall the units for the quantities involved in both expressions: 1. Energy (E) is measured in joules (J) 2. Time (t) is measured in seconds (s) 3. Position (x) is measured in meters (m) 4. Mass (m) is measured in kilograms (kg) 5. Speed (v) is measured in meters per second (m/s) 6. Planck's constant (h) has units of joule-seconds (J·s) Now that we know the units of each quantity, we can check the units of each expression.

In the first expression, we have: \[ \Delta E \cdot \Delta t \geq \frac{h}{4 \pi} \] Analyzing the units, we can see that: - \(\Delta E\) has units of joules (J) - \(\Delta t\) has units of seconds (s) - \(h\) has units of joule-seconds (J·s) So the expression becomes: \[ J \cdot s \geq \frac{J \cdot s}{1} \] Which simplifies to: \[ J \cdot s \geq J \cdot s \] The units on both sides of the inequality are the same, so we're good to go.

In the second expression, we have: \[ \Delta x \cdot \Delta(m v) \geq \frac{h}{4 \pi} \] Analyzing the units, we can see that: - \(\Delta x\) has units of meters (m) - \(\Delta(m v)\) has units of kg·(m/s) (since mass has units of kg and speed has units of m/s) - \(h\) has units of joule-seconds (J·s) Multiplying the units for \(\Delta x\) and \(\Delta(m v)\), we get: \[ \text{m} \cdot \text{kg} \cdot \frac{\text{m}}{\text{s}} \] Now we need to find a way to express this in joules and seconds. Since 1 joule is equal to 1 kg·m²/s² (J = kg·m²/s²), we can rewrite the above expression as: \[ \frac{\text{J} \cdot \text{s}}{\text{m}} \] Now that we have a common unit (joules) for both expressions, we can rewrite the second expression: \[ \frac{J \cdot s}{m} \geq \frac{J \cdot s}{1} \] Multiplying both sides of the inequality by m, we get: \[ J \cdot s \geq J \cdot s \]

The units for the first expression are: \[ J \cdot s \geq J \cdot s \] The units for the second expression are: \[ J \cdot s \geq J \cdot s \] As we can see, the units of both expressions are the same, confirming that both forms of the Heisenberg uncertainty principle have the same units.