An electron can be in one of two potential wells that are so...

An electron can be in one of two potential wells that are so close that it can 'tunnel' from one to the other (see $\S 5.2$ for a description of quantum-mechanical tunnelling). Its state vector can be written $$ |\psi\rangle=a|A\rangle+b|B\rangle $$ where $|A\rangle$ is the state of being in the first well and $|B\rangle$ is the state of being in the second well and all kets are correctly normalised. What is the probability of finding the particle in the first well given that: (a) $a=\mathrm{i} / 2 ;$ (b) $b=\mathrm{e}^{\mathrm{i} \pi} ;$ (c) $b=\frac{1}{\mathrm{v}}+\mathrm{i} / \sqrt{2} ?$

An electron can be in one of two potential wells that are so close that it can 'tunnel' from one to the other (see $\S 5.2$ for a description of quantum-mechanical tunnelling). Its state vector can be written
$$
|\psi\rangle=a|A\rangle+b|B\rangle
$$
where $|A\rangle$ is the state of being in the first well and $|B\rangle$ is the state of being in the second well and all kets are correctly normalised. What is the probability of finding the particle in the first well given that: (a) $a=\mathrm{i} / 2 ;$ (b) $b=\mathrm{e}^{\mathrm{i} \pi} ;$ (c) $b=\frac{1}{\mathrm{v}}+\mathrm{i} / \sqrt{2} ?$

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Key Concepts

Complex Numbers in Quantum Mechanics

In quantum mechanics, amplitudes are generally complex numbers. The phase and magnitude of these complex numbers play a critical role in interference and other quantum phenomena, and the modulus squared operation is used to extract real, physical probabilities from these complex numbers.

Normalization in Quantum Mechanics

Normalization ensures that the total probability of all outcomes is equal to one. It requires that the sum of the squared moduli of all probability amplitudes in a state vector equals one, which is a key condition for a valid quantum state.

Probability Amplitude

Probability amplitudes are the complex coefficients in the state vector that multiply the basis states. The square of the absolute value (modulus) of each amplitude gives the probability of observing the corresponding measurement outcome when the system is measured, according to the Born rule.

Quantum State Vector

A quantum state vector is a mathematical representation of a quantum system, typically expressed in a Hilbert space. It encapsulates all the information about the system and is usually written as a linear combination of basis states, where each basis state corresponds to a possible measurement outcome.

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