Probability Amplitudes: The Core of Quantum Prediction

Table of Contents

1. Introduction
2. What Are Probability Amplitudes?
3. Mathematical Definition
4. Born Rule and Measurement
5. Superposition Principle
6. Inner Products and Transition Amplitudes
7. Amplitudes in Different Representations
8. Probability Densities and Continuum States
9. Normalization and Orthogonality
10. Interference and Phase Information
11. Two-State Systems and Qubits
12. Complex Numbers and Physical Implications
13. Amplitude vs Probability
14. Amplitudes in Path Integral Formulation
15. Applications in Quantum Computing and Optics
16. Conclusion

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1. Introduction

Probability amplitudes are the fundamental quantities in quantum mechanics from which all measurable predictions are derived. They form the backbone of quantum theory’s probabilistic nature, encapsulating both magnitude and phase — crucial for understanding interference and superposition.

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2. What Are Probability Amplitudes?

In quantum mechanics, a probability amplitude is a complex number whose modulus squared gives the probability of finding a system in a particular state. Unlike classical probabilities, these amplitudes can interfere constructively or destructively.

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3. Mathematical Definition

If \( |\psi\rangle \) is the quantum state, and \( |a\rangle \) is an eigenstate of some observable, then the probability amplitude to find \( a \) is:

\[
\langle a | \psi \rangle
\]

The probability of outcome \( a \) is:

\[
P(a) = |\langle a | \psi \rangle|^2
\]

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4. Born Rule and Measurement

Formulated by Max Born:

\[
P(a) = |\langle a | \psi \rangle|^2
\]

• Core rule of quantum mechanics
• Gives the link between theory and experiment
• Measurement collapses the state to \( |a\rangle \)

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5. Superposition Principle

If a system is in a superposition:

\[
|\psi\rangle = c_1 |a_1\rangle + c_2 |a_2\rangle
\]

Then \( c_1 = \langle a_1 | \psi \rangle \), and \( |c_1|^2 \) gives the probability of finding \( a_1 \). The interference of these amplitudes leads to non-classical phenomena.

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6. Inner Products and Transition Amplitudes

Transition from state \( |\phi\rangle \) to \( |\psi\rangle \):

\[
A = \langle \psi | \phi \rangle
\]

Transition probability:

\[
P = |\langle \psi | \phi \rangle|^2
\]

Used in time evolution and scattering processes.

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7. Amplitudes in Different Representations

Position basis:

\[
\psi(x) = \langle x | \psi \rangle
\]

Momentum basis:

\[
\tilde{\psi}(p) = \langle p | \psi \rangle
\]

Amplitudes can be transformed between bases via Fourier transforms.

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8. Probability Densities and Continuum States

In continuous systems:

• \( |\psi(x)|^2 \ dx \) gives the probability of finding particle between \( x \) and \( x + dx \)
• Normalization:
  \[
  \int_{-\infty}^\infty |\psi(x)|^2 dx = 1
  \]

Amplitudes can be distributions in infinite-dimensional Hilbert spaces.

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9. Normalization and Orthogonality

For state \( |\psi\rangle \):

• Normalization:
  \[
  \langle \psi | \psi \rangle = 1
  \]
• Orthonormal basis \( \{ |a_i\rangle \} \):
  \[
  \langle a_i | a_j \rangle = \delta_{ij}
  \]

Completeness ensures total probability is conserved.

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10. Interference and Phase Information

Amplitudes encode phase. For two paths with amplitudes \( A_1 \) and \( A_2 \):

\[
P = |A_1 + A_2|^2 = |A_1|^2 + |A_2|^2 + 2 \text{Re}(A_1^* A_2)
\]

Interference term depends on relative phase — crucial in double-slit experiments and quantum optics.

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11. Two-State Systems and Qubits

Qubit state:

\[
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle
\]

• \( \alpha \) and \( \beta \) are probability amplitudes
• Measurement yields 0 with probability \( |\alpha|^2 \), 1 with \( |\beta|^2 \)

Quantum computation relies on manipulating these amplitudes through unitary operations.

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12. Complex Numbers and Physical Implications

Complex amplitudes allow:

• Cancellation and enhancement
• Description of oscillatory behavior
• Conservation via unitary evolution

Phases are physically meaningful in interference and entanglement.

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13. Amplitude vs Probability

Aspect	Amplitude \( \langle a	\psi \rangle \)	Probability \(	\langle a	\psi \rangle	^2 \)
Type	Complex number	Real number
Contains phase?	Yes	No
Linear?	Yes	No (quadratic)
Interference	Possible	No (phase removed)

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14. Amplitudes in Path Integral Formulation

In Feynman’s path integral approach:

\[
\langle x_f, t_f | x_i, t_i \rangle = \int \mathcal{D}[x(t)] \, e^{iS[x]/\hbar}
\]

Each path contributes a complex amplitude, and summing them yields total probability amplitude.

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15. Applications in Quantum Computing and Optics

• Quantum algorithms manipulate amplitudes (e.g., Grover’s search)
• Interference used for speed-up
• Quantum teleportation and communication rely on precise amplitude control
• Optical interferometers measure tiny phase differences

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16. Conclusion

Probability amplitudes are the heart of quantum mechanics. They unify wave and particle behaviors, encapsulate probabilities and interference, and define the quantum logic that governs everything from atomic transitions to quantum computers. Understanding and controlling amplitudes is the key to unlocking the power and mystery of the quantum world.

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