Answer
See the explanation
Work Step by Step
To show that the units for both forms are the same, let's first analyze the units for each term:
1. For the left side of the equation \(\Delta E \cdot \Delta t\):
- \(\Delta E\) represents energy, which has units of joules (J).
- \(\Delta t\) represents time, which has units of seconds (s).
- Therefore, the units for \(\Delta E \cdot \Delta t\) are joules times seconds (J⋅s).
2. For the right side of the equation \(\frac{h}{4 \pi}\):
- \(h\) is the Planck constant, which has units of joule-seconds (J⋅s).
- \(\pi\) is a dimensionless constant.
- Therefore, the units for \(\frac{h}{4 \pi}\) are also joules times seconds (J⋅s).
Now, let's analyze the units for the second form:
1. For the left side of the equation \(\Delta x \cdot \Delta(mv)\):
- \(\Delta x\) represents displacement, which has units of meters (m).
- \(\Delta (mv)\) represents change in momentum, which has units of kilogram-meters per second (kg⋅m/s).
- Therefore, the units for \(\Delta x \cdot \Delta (mv)\) are meters times kilogram-meters per second (m⋅kg⋅m/s), which simplifies to \(m^2 \cdot kg/s\).
2. For the right side of the equation \(\frac{h}{4 \pi}\):
- As mentioned earlier, the units are joules times seconds (J⋅s).
To show that these units are the same, we need to demonstrate that \(m^2 \cdot kg/s\) is equivalent to \(J \cdot s\).
Recall that the units of momentum are \(kg \cdot m/s\), and since \(1 \, J = 1 \, kg \cdot m^2/s^2\), we can rewrite \(J\) as \(kg \cdot m^2/s^2\).
So, \(J \cdot s = (kg \cdot m^2/s^2) \cdot s = kg \cdot m^2/s\), which matches the units of \(\Delta x \cdot \Delta(mv)\).
Therefore, the units for both forms are indeed the same, confirming the equivalence of the two expressions of the Heisenberg uncertainty principle.
