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Some usual calculations of the Bohr's radius (see 2.5 at Feynman's here, or this text here) starts by defining a radius $a$ ( most probable radius ? average radius ? ). Next, from Heisenberg's uncertainty principle:

$$\Delta p \Delta a \ge h/2$$

it is said:

$$p = h/a$$.

where $p$ is the electron momentum.

Remainder of the calculation is expression of the total electron energy and minimization.

This step from uncertainty to momentum is confusing to me: we pass from inequality to equality, from margins ($\Delta$) to concrete values of $p$ and $a$.

Any hint to understand this step? Thanks.

Addendum:

This is another similar example, from "Quantum Mechanics", Nouredine Zettili, example 1.6:

Estimate the uncertainty in the position of (a) a neutron moving at $5 \cdot 10^6 m/s$ ... Solution: Using (1.57), we can write the position uncertainty as $$\Delta x = \frac{\hbar}{2 \Delta p} = \frac{\hbar}{2 m v } = ... $$

again, $\Delta p$, an interval, is converted to the absolute value $p=mv$.

If something is in an interval $p \in [p_{min},p_{max}]$ we can say $p=p_{avg}\pm \Delta p$ where $p_{avg}=\frac{p_{max}+p_{min}}{2}$ and $\Delta p=\frac{p_{max}-p_{min}}{2}$, but these texts seems to assume $p_{min}=0$ and $\frac{p_{max}}{2}=mv=\Delta p$ or something similar.

Some usual calculations of the Bohr's radius (see 2.5 at Feynman's here, or this text here) starts by defining a radius $a$ ( most probable radius ? average radius ? ). Next, from Heisenberg's uncertainty principle:

$$\Delta p \Delta a \ge h/2$$

it is said:

$$p = h/a$$.

where $p$ is the electron momentum.

Remainder of the calculation is expression of the total electron energy and minimization.

This step from uncertainty to momentum is confusing to me: we pass from inequality to equality, from margins ($\Delta$) to concrete values of $p$ and $a$.

Any hint to understand this step? Thanks.

Addendum:

This is another similar example, from "Quantum Mechanics", Nouredine Zettili, example 1.6:

Estimate the uncertainty in the position of (a) a neutron moving at $5 \cdot 10^6 m/s$ ... Solution: Using (1.57), we can write the position uncertainty as $$\Delta x = \frac{\hbar}{2 \Delta p} = \frac{\hbar}{2 m v } = ... $$

again, $\Delta p$, an interval, is converted to the absolute value $p=mv$.

If something is in an interval $p \in [p_{min},p_{max}]$ we can say $p=p_{avg}\pm \Delta p$ where $p_{avg}=\frac{p_{max}+p_{min}}{2}$ and $\Delta p=\frac{p_{max}-p_{min}}{2}$, but these texts seems to assume $p_{min}=0$ and $\frac{p_{max}}{2}=mv=\Delta p$ or something similar.

In other words, when in this example we give a known value for the speed, we are fixing the momentum without any indetermination, thus, uncertainly in position should be infinite.

Some usual calculations of the Bohr's radius (see 2.5 at Feynman's here, or this text here) starts by defining a radius $a$ ( most probable radius ? average radius ? ). Next, from Heisenberg's uncertainty principle:

$$\Delta p \Delta a \ge h/2$$

it is said:

$$p = h/a$$.

where $p$ is the electron momentum.

Remainder of the calculation is expression of the total electron energy and minimization.

This step from uncertainty to momentum is confusing to me: we pass from inequality to equality, from margins ($\Delta$) to concrete values of $p$ and $a$.

Any hint to understand this step? Thanks.

Addendum:

This is another similar example, from "Quantum Mechanics", Nouredine Zettili, example 1.6:

Estimate the uncertainty in the position of (a) a neutron moving at $5 \cdot 10^6 m/s$ ... Solution: Using (1.57), we can write the position uncertainty as $$\Delta x = \frac{\hbar}{2 \Delta p} = \frac{\hbar}{2 m v } = ... $$

again, $\Delta p$, an interval, is converted to the absolute value $p=mv$.

If something is in an interval $p \in [p_{min},p_{max}]$ we can say $p=p_{avg}\pm \Delta p$ where $p_{avg}=\frac{p_{max}+p_{min}}{2}$ and $\Delta p=\frac{p_{max}-p_{min}}{2}$, but these texts seems to assume $p_{min}=0$ and $2\,p_{max}=p=\Delta p$ or something similar.

Some usual calculations of the Bohr's radius (see 2.5 at Feynman's here, or this text here) starts by defining a radius $a$ ( most probable radius ? average radius ? ). Next, from Heisenberg's uncertainty principle:

$$\Delta p \Delta a \ge h/2$$

it is said:

$$p = h/a$$.

where $p$ is the electron momentum.

Remainder of the calculation is expression of the total electron energy and minimization.

This step from uncertainty to momentum is confusing to me: we pass from inequality to equality, from margins ($\Delta$) to concrete values of $p$ and $a$.

Any hint to understand this step? Thanks.

Addendum:

This is another similar example, from "Quantum Mechanics", Nouredine Zettili, example 1.6:

Estimate the uncertainty in the position of (a) a neutron moving at $5 \cdot 10^6 m/s$ ... Solution: Using (1.57), we can write the position uncertainty as $$\Delta x = \frac{\hbar}{2 \Delta p} = \frac{\hbar}{2 m v } = ... $$

again, $\Delta p$, an interval, is converted to the absolute value $p=mv$.

If something is in an interval $p \in [p_{min},p_{max}]$ we can say $p=p_{avg}\pm \Delta p$ where $p_{avg}=\frac{p_{max}+p_{min}}{2}$ and $\Delta p=\frac{p_{max}-p_{min}}{2}$, but these texts seems to assume $p_{min}=0$ and $\frac{p_{max}}{2}=mv=\Delta p$ or something similar.

Some usual calculations of the Bohr's radius (see 2.5 at Feynman's here, or this text here) starts by defining a radius $a$ ( most probable radius ? average radius ? ). Next, from Heisenberg's uncertainty principle:

$$\Delta p \Delta a \ge h/2$$

it is said:

$$p = h/a$$.

where $p$ is the electron momentum.

Remainder of the calculation is expression of the total electron energy and minimization.

This step from uncertainty to momentum is confusing to me: we pass from inequality to equality, from margins ($\Delta$) to concrete values of $p$ and $a$.

Any hint to understand this step? Thanks.

Some usual calculations of the Bohr's radius (see 2.5 at Feynman's here, or this text here) starts by defining a radius $a$ ( most probable radius ? average radius ? ). Next, from Heisenberg's uncertainty principle:

$$\Delta p \Delta a \ge h/2$$

it is said:

$$p = h/a$$.

where $p$ is the electron momentum.

Remainder of the calculation is expression of the total electron energy and minimization.

This step from uncertainty to momentum is confusing to me: we pass from inequality to equality, from margins ($\Delta$) to concrete values of $p$ and $a$.

Any hint to understand this step? Thanks.

Addendum:

This is another similar example, from "Quantum Mechanics", Nouredine Zettili, example 1.6:

Estimate the uncertainty in the position of (a) a neutron moving at $5 \cdot 10^6 m/s$ ... Solution: Using (1.57), we can write the position uncertainty as $$\Delta x = \frac{\hbar}{2 \Delta p} = \frac{\hbar}{2 m v } = ... $$

again, $\Delta p$, an interval, is converted to the absolute value $p=mv$.

If something is in an interval $p \in [p_{min},p_{max}]$ we can say $p=p_{avg}\pm \Delta p$ where $p_{avg}=\frac{p_{max}+p_{min}}{2}$ and $\Delta p=\frac{p_{max}-p_{min}}{2}$, but these texts seems to assume $p_{min}=0$ and $2\,p_{max}=p=\Delta p$ or something similar.