Quantum Inner Product Calculator

Quantum Inner Product Formula:

\[ \langle\psi|\phi\rangle = \sum_i c_i^* d_i \quad \text{where} \quad |\psi\rangle = \sum_i c_i |i\rangle, \quad |\phi\rangle = \sum_i d_i |i\rangle \]

Inner Product Result (⟨ψ|φ⟩):

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1. What is the Quantum Inner Product?

The quantum inner product (⟨ψ|φ⟩) is a fundamental operation in quantum mechanics that measures the overlap between two quantum states. It's used to calculate probabilities, determine orthogonality, and perform state measurements.

2. How Does the Calculator Work?

The calculator uses the quantum inner product formula:

\[ \langle\psi|\phi\rangle = \sum_i c_i^* d_i \]

Where:

\( c_i \) — Complex coefficients of state |ψ⟩
\( d_i \) — Complex coefficients of state |φ⟩
\( c_i^* \) — Complex conjugate of \( c_i \)

Explanation: The calculator computes the sum of products of complex-conjugated coefficients from the first state with coefficients from the second state.

3. Importance of Inner Product in Quantum Mechanics

Details: The inner product is essential for calculating transition probabilities between states, testing orthogonality, and performing quantum measurements. It's the foundation of the Born rule in quantum mechanics.

4. Using the Calculator

Tips: Enter coefficients as comma-separated complex numbers (e.g., "1, 0.5i, -0.3" or "1+i, 2-0.5i"). Both states must have the same number of coefficients.

5. Frequently Asked Questions (FAQ)

Q1: What does a zero inner product mean?
A: A zero inner product indicates that the two states are orthogonal (perpendicular in Hilbert space).

Q2: How do I represent complex numbers?
A: Use 'i' for imaginary unit (e.g., "1+2i", "0.5i", "-3-0.5i"). No spaces within a number.

Q3: What's the physical interpretation?
A: The modulus squared of the inner product gives the probability of measuring |φ⟩ when the system is in state |ψ⟩.

Q4: What about normalization?
A: This calculator doesn't normalize inputs. For probabilities, states should be normalized (⟨ψ|ψ⟩ = 1).

Q5: Can I use this for multi-qubit systems?
A: Yes, as long as you provide all coefficients in the computational basis (may need 2^n coefficients for n qubits).