12.2.2 Exploring Heisenberg uncertainty principle: <latex>\Delta x \Delta p \geq \frac{\hbar}{2}</latex>

• The Heisenberg uncertainty principle states that there is a fundamental limit to how precisely we can know both the position and momentum of a particle simultaneously.
• What happens to the uncertainty in momentum if the uncertainty in position decreases?

  It increases
• Match the variable with its definition:

  $\Delta x$ ↔️ Uncertainty in position

  $\Delta p$ ↔️ Uncertainty in momentum

  $\hbar$ ↔️ Planck's reduced constant
• Planck's reduced constant is defined as $\hbar =$$\frac{h}{2\pi}$, where $h$ is Planck's constant.
• What is the approximate value of Planck's reduced constant?

  1.054 × 10⁻³⁴ J⋅s
• What is the mathematical expression for the Heisenberg uncertainty principle?

  $\Delta x \Delta p \geq \frac{\hbar}{2}$
• The more precisely we determine a particle's position, the more precisely we can know its momentum.

  False
• For macroscopic objects, the uncertainties in position and momentum are negligible
• The value of Planck's constant is approximately 6.626 × 10⁻³⁴ J⋅s.


  True
• Arrange the following pairs of uncertainties in position and momentum based on their relationship:

  1️⃣ Small uncertainty in position, large uncertainty in momentum

  2️⃣ Large uncertainty in position, small uncertainty in momentum
• The Planck constant sets the lower limit for the uncertainties in position and momentum.


  True
• For macroscopic objects, the uncertainties in position and momentum are negligible.
• In the Heisenberg uncertainty principle, Δx represents the uncertainty in position.


  True
• What does the Heisenberg uncertainty principle state about the simultaneous knowledge of position and momentum?

  Impossible to know exactly
• Planck's constant sets the scale for the uncertainties in the Heisenberg uncertainty principle, which are significant for quantum particles.
• The Planck constant sets the scale for the uncertainty relationship in the Heisenberg uncertainty principle.


  True
• For macroscopic objects, the uncertainties in position and momentum are considered negligible.
• The Heisenberg uncertainty principle implies that the more precisely we know a particle's momentum, the less precisely we know its position.

  True
• Planck's constant has a value of approximately 6.626 × 10⁻³⁴ J⋅s.


  True
• The Heisenberg uncertainty principle is expressed as $\Delta x \Delta p \geq \frac{\hbar}{2}$.


  True
• Planck's reduced constant is denoted by the symbol ℏ
• Match the uncertainty in position with its corresponding uncertainty in momentum:

  More precisely known position ↔️ Less precisely known momentum

  Less precisely known position ↔️ More precisely known momentum
• What is the definition of Planck's reduced constant in terms of Planck's constant?

  $\hbar =$$\frac{h}{2\pi}$
• The value of Planck's reduced constant is approximately 1.054 × 10⁻³⁴ J⋅s.
• The product of the uncertainties in position and momentum must always be greater than or equal to \hbar / 2.
• What does the Heisenberg uncertainty principle state in simple terms?

  Fundamental limit on precision
• What is the mathematical expression for the Heisenberg uncertainty principle?

  $\Delta x \Delta p \geq \frac{\hbar}{2}$
• The Planck constant ℏ sets the scale for the uncertainty relationship in the Heisenberg uncertainty principle.
• The Heisenberg uncertainty principle is a result of imperfections in measurement tools.

  False
• What is the role of Planck's reduced constant ℏ in the Heisenberg uncertainty principle?

  Sets the scale
• Match the constant with its definition:

  Planck's Constant ($h$) ↔️ 6.626 × 10⁻³⁴ J⋅s

  Planck's Reduced Constant ($\hbar$) ↔️ 1.054 × 10⁻³⁴ J⋅s
• The inequality $\Delta x \Delta p \geq \frac{\hbar}{2}$ in the Heisenberg uncertainty principle indicates that the product of the uncertainties must always be greater than or equal to half the Planck constant.
• What does the Heisenberg uncertainty principle state about the relationship between uncertainty in position and momentum?

  $\Delta x \Delta p \geq \frac{\hbar}{2}$
• The Heisenberg uncertainty principle states that both position and momentum of a particle can be known with perfect precision simultaneously.

  False
• Match the concept with its description:

  Simultaneous Measurements ↔️ Impossibility of knowing both position and momentum with perfect precision

  Inverse Relationship ↔️ Higher accuracy in knowing position results in less accuracy in knowing momentum, and vice versa

  Quantum Phenomena ↔️ Essential in understanding quantum tunneling and the spread of wave functions

  Planck Constant ↔️ Sets the scale for uncertainties in quantum mechanics
• The Heisenberg uncertainty principle applies equally to macroscopic and quantum objects.

  False
• The Planck constant $\hbar$ sets the scale for the uncertainties in the Heisenberg uncertainty principle, which are negligible for macroscopic objects but significant for quantum particles.
• What role does the Planck constant play in the Heisenberg uncertainty principle?

  Sets the scale for uncertainties
• Planck's reduced constant is also known as Dirac's constant.
• The value of Planck's reduced constant ℏ is approximately $1.054 × 10^{ - 34} \, \text{J⋅s}$.


  True