Uncertainty
1. Classical Uncertainty | Response time - bandwidth relation (2 pt).
A certain data-cable transfers information from a source (S) to a receiver (R) in the form
of brief electric pulses at a rate of 100,000 pulses/s.
(a) What is the maximum time-span of a single pulse such that subsequent pulses do not
overlap?
(b) To pick up the signal, what range of pulse frequencies (fmin, fmax) must the receiving
equipment be sensitive to?
2. Uncertainty Principle (E,t) (1 pt).
If an excited state of an atom is known to have a lifetime of 10−9 s, what is the uncertainty
in the energy of photons emitted by such atoms in the spontaneous decay to the ground
state? Assume that ℏ ≈ 10−34 J·s.
3. Position-Momentum Uncertainty Principle (2 pt).
An entomologist is studying a small beetle. She records the beetle’s weight to be 1.0 mg
and its diameter to be 3 mm. At one point she is using a magnifier and observes the beetle
to be stationary with an uncertainty of 10−2 mm. How fast might the beetle actually be
moving?

Let's tackle each part of your question step by step.

### 1. Classical Uncertainty | Response time - bandwidth relation (2 pt)

#### (a) Maximum time-span of a single pulse

The rate of pulses is given as 100,000 pulses/s. The time period (T) for one pulse can be calculated as the inverse of the pulse rate:

\[
T = \frac{1}{\text{Rate}} = \frac{1}{100,000 \, \text{pulses/s}} = 10^{-5} \, \text{s} = 10 \, \mu s
\]

This means the maximum time-span for a single pulse, such that subsequent pulses do not overlap, is **10 microseconds**.

#### (b) Range of pulse frequencies (f_min, f_max)

The frequency of the pulses corresponds directly to their rate. The minimum frequency (f_min) is essentially the frequency at which a single pulse occurs, while the maximum frequency (f_max) would correspond to the highest rate that can still be recognized by the receiving equipment.

Given:
- \( f = \text{Rate} = 100,000 \, \text{pulses/s} \)

Thus,
- \( f_{min} = 100,000 \, \text{Hz} \)
- \( f_{max} \) would be determined by the bandwidth limitations of the receiver but can be approximated by the Nyquist rate, which is typically twice the maximum frequency for accurate signal representation. 

For simplicity, without additional details on bandwidth, we can consider \( f_{max} \) to be around twice the pulse frequency:
- \( f_{max} \approx 200,000 \, \text{Hz} \)

### 2. Uncertainty Principle (E,t) (1 pt)

The uncertainty in energy (\( \Delta E \)) can be calculated using the uncertainty principle:

\[
\Delta E \cdot \Delta t \geq \frac{\hbar}{2}
\]

Given that the lifetime of the state (\( \Delta t \)) is \( 10^{-9} \, \text{s} \) and \( \hbar \approx 10^{-34} \, \text{J·s} \):

\[
\Delta E \geq \frac{\hbar}{2 \Delta t} = \frac{10^{-34} \, \text{J·s}}{2 \times 10^{-9} \, \text{s}} = 5 \times 10^{-26} \, \text{J}
\]

Thus, the uncertainty in the energy of photons emitted by the atoms is approximately **\( 5 \times 10^{-26} \, \text{J} \)**.

### 3. Position-Momentum Uncertainty Principle (2 pt)

Using the position-momentum uncertainty principle:

\[
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}
\]

Where:
- \( \Delta x \) (uncertainty in position) = \( 10^{-2} \, \text{mm} = 10^{-5} \, \text{m} \)
- The mass of the beetle (m) = \( 1.0 \, \text{mg} = 1.0 \times 10^{-6} \, \text{kg} \)

The momentum (\( p \)) is related to velocity (\( v \)) by:

\[
p = mv
\]

Thus:

\[
\Delta p = m \Delta v
\]

Substituting into the uncertainty principle:

\[
10^{-5} \cdot m \Delta v \geq \frac{10^{-34}}{2}
\]

Calculating \( m \Delta v \):

\[
\Delta v \geq \frac{10^{-34}}{2 \cdot 10^{-5} \cdot 10^{-6}} = \frac{10^{-34}}{10^{-11}} = 10^{-23} \, \text{m/s}
\]

Thus, the beetle might be moving at a speed uncertainty of approximately **\( 10^{-23} \, \text{m/s} \)**.

### Summary of Results:
1. (a) Maximum time-span of a pulse: **10 µs**
   (b) Pulse frequency range: \( f_{min} = 100,000 \, \text{Hz}, f_{max} \approx 200,000 \, \text{Hz} \)
2. Uncertainty in energy: **\( 5 \times 10^{-26} \, \text{J} \)**
3. Speed uncertainty for the beetle: **\( 10^{-23} \, \text{m/s} \)**

Would you like any further details or clarifications? Here are some questions to consider:
1. How does the bandwidth of a receiver affect its ability to pick up signals?
2. Can you explain more about the implications of the uncertainty principle in quantum mechanics?
3. What practical applications exist for measuring uncertainty in energy?
4. How does the uncertainty in position relate to the observable behavior of particles?
5. What are some other scenarios where these principles might apply?

**Tip:** When dealing with uncertainties, always ensure units are consistent throughout calculations to avoid errors.

Uncertainty 1. Classical Uncertainty | Response time - bandwidth relation (2 pt). A certain data-cable transfers information from a source (S) to a receiver (R) in the form of brief electric pulses at a rate of 100,000 pulses/s. (a) What is the maximum time-span of a single pulse such that subsequent pulses do not overlap? (b) To pick up the signal, what range of pulse frequencies (fmin, fmax) must the receiving equipment be sensitive to? 2. Uncertainty Principle (E,t) (1 pt). If an excited state of an atom is known to have a lifetime of 10−9 s, what is the uncertainty in the energy of photons emitted by such atoms in the spontaneous decay to the ground state? Assume that ℏ ≈ 10−34 J·s. 3. Position-Momentum Uncertainty Principle (2 pt). An entomologist is studying a small beetle. She records the beetle’s weight to be 1.0 mg and its diameter to be 3 mm. At one point she is using a magnifier and observes the beetle to be stationary with an uncertainty of 10−2 mm. How fast might the beetle actually be moving?

Explore the intricacies of uncertainty principles in physics, including classical pulse timing, energy uncertainty due to lifetime, and position-momentum relationships, with detailed examples and solutions. Suitable for advanced high school students.

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