Normalization Probability amplitude
in example above, measurement must give either | h ⟩ or | v ⟩, total probability of measuring | h ⟩ or | v ⟩ must 1. leads constraint α + β = 1; more sum of squared moduli of probability amplitudes of possible states equal one. if understand possible states orthonormal basis, makes sense in discrete case, condition same norm-1 condition explained above.
one can divide non-zero element of hilbert space norm , obtain normalized state vector. not every wave function belongs hilbert space l(x), though. wave functions fulfill constraint called normalizable.
the schrödinger wave equation, describing states of quantum particles, has solutions describe system , determine precisely how state changes time. suppose wavefunction ψ0(x, t) solution of wave equation, giving description of particle (position x, time t). if wavefunction square integrable, i.e.
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{\displaystyle \int _{\mathbf {r} ^{n}}|\psi _{0}(\mathbf {x} ,t_{0})|^{2}\,\mathrm {d\mathbf {x} } =a^{2}<\infty }
for t0, ψ = ψ0/a called normalized wavefunction. under standard copenhagen interpretation, normalized wavefunction gives probability amplitudes position of particle. hence, @ given time t0, ρ(x) = |ψ(x, t0)| probability density function of particle s position. probability particle in volume v @ t0 is
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{\displaystyle \mathbf {p} (v)=\int _{v}\rho (\mathbf {x} )\,\mathrm {d\mathbf {x} } =\int _{v}|\psi (\mathbf {x} ,t_{0})|^{2}\,\mathrm {d\mathbf {x} } .}
note if solution ψ0 wave equation normalisable @ time t0, ψ defined above normalised, that
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{\displaystyle \rho _{t}(\mathbf {x} )=\left|\psi (\mathbf {x} ,t)\right|^{2}=\left|{\frac {\psi _{0}(\mathbf {x} ,t)}{a}}\right|^{2}}
is probability density function t. key understanding importance of interpretation, because given particle s constant mass, initial ψ(x, 0) , potential, schrödinger equation determines subsequent wavefunction, , above gives probabilities of locations of particle @ subsequent times.
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