1.0×10⁶ atoms are excited to an upper energy level at t = 0 s. At the end of 20 ns, 90% of these atoms have undergone a quantum jump to the ground state. What is the lifetime of the excited state?

검증된 단계별 안내

Understand the problem: The lifetime of an excited state is related to the decay of the number of excited atoms over time. This decay follows an exponential law: \( N(t) = N_0 e^{-t/\tau} \), where \( N(t) \) is the number of atoms remaining in the excited state at time \( t \), \( N_0 \) is the initial number of excited atoms, and \( \tau \) is the lifetime of the excited state.

Identify the given values: \( N_0 = 1.0 \times 10^6 \) atoms, \( t = 20 \; \text{ns} = 20 \times 10^{-9} \; \text{s} \), and \( N(t) = 10\% \times N_0 = 0.1 \times N_0 \) (since 90% of the atoms have decayed).

Substitute the known values into the exponential decay formula: \( 0.1 N_0 = N_0 e^{-t/\tau} \). Simplify by dividing through by \( N_0 \), which gives \( 0.1 = e^{-t/\tau} \).

Take the natural logarithm of both sides to solve for \( \tau \): \( \ln(0.1) = -t/\tau \). Rearrange to isolate \( \tau \): \( \tau = -t / \ln(0.1) \).

Substitute \( t = 20 \times 10^{-9} \; \text{s} \) and calculate \( \ln(0.1) \) (which is a constant value). This will give you the lifetime \( \tau \) of the excited state in seconds.

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주요 개념

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Quantum State Lifetime

The lifetime of an excited state refers to the average time an atom remains in an excited energy level before transitioning to a lower energy state. This process is probabilistic, meaning that while some atoms may transition quickly, others may take longer. The lifetime is a crucial parameter in understanding the dynamics of atomic transitions and is often denoted by the symbol τ (tau).

Exponential Decay

In quantum mechanics, the population of excited atoms decreases over time according to an exponential decay law. This means that the number of atoms remaining in the excited state decreases by a constant fraction in equal time intervals. The relationship can be expressed mathematically as N(t) = N0 * e^(-t/τ), where N0 is the initial number of excited atoms, and τ is the lifetime of the excited state.

Percentage Decay

Percentage decay refers to the fraction of a population that has transitioned from an excited state to a ground state over a specified time period. In this scenario, 90% of the excited atoms have decayed, indicating that only 10% remain excited. This information is essential for calculating the lifetime of the excited state using the exponential decay formula, as it provides a direct relationship between the remaining population and the time elapsed.