States as Vectors: Dirac Notation and the State Space
A quantum state is a vector in a complex space carrying an inner product, and the wavefunction is only its components in one particular basis. Bra-ket notation writes the same state in position, momentum or energy language without changing the state itself.
What a learner can do afterwards
- Writes a state as a superposition of basis kets and reads off the amplitudes
- Converts between a ket and the wavefunction that represents it in a chosen basis
- Uses an inner product to compute an overlap and says what that number predicts
1 · Read
A quantum state is a vector in a complex space carrying an inner product. Bra-ket notation writes the state itself as a ket, independent of any coordinate choice. The wavefunction is only its components in one basis, such as position. Position, momentum, and energy are three bases for the same space, each an orthonormal set spanning it.
Write a state as a superposition of two basis kets with amplitudes 0.6 and 0.8. Each amplitude is the complex component along its basis ket, and its squared size is the probability: 0.36 for the first outcome and 0.64 for the second. They sum to 1, so the state vector must have length 1. Never confuse the amplitude with the probability.
The inner product of two states is their bracket: a complex number whose size measures their overlap. One bracket against a basis ket reads off an amplitude, and orthonormal bases keep expansions clean. Changing basis changes nothing physical: it is the same vector in a new language, like coordinates in a rotated frame. Every prediction stays identical.
Keep three distinctions straight. The ket is the state itself; the wavefunction is its position components. The amplitude is the component; its squared size is the probability. The basis is your descriptive choice; the physics is basis independent. Work through one state in two bases and the notation stops feeling mysterious.
States are vectors, wavefunctions are components, and squared amplitudes give the odds.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.