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Abstract quantum wave interference patterns representing quantum mechanics

Hilbert space and state vectors

PHYS 410 · Quantum Formalism

Quantum states live in Hilbert space, a complex vector space with an inner product. This lesson connects wavefunctions, vectors, bases, superposition, and normalization.

Key equations

|psi anglepsi(x)=langle x|psi anglelangle phi|psi angle||psi||=sqrt{langlepsi|psi angle}langlepsi|psi angle=1|psi angle=a|psi_1 angle+b|psi_2 anglelangle m|n angle=delta_{mn}|psi angle=sum_n c_n|n anglec_n=langle n|psi anglesum_n |c_n|^2=1|psi angle=int psi(x)|x angle dxlangle x|x' angle=delta(x-x')|psi angle ightarrow e^{ialpha}|psi angle

Learning objectives

  • Define a quantum state vector.
  • Explain Hilbert space conceptually.
  • Relate wavefunctions to basis representations.
  • Use orthonormal basis expansions.
  • Distinguish global and relative phase.

From wavefunctions to state vectors

Wave mechanics describes a particle using a wavefunction psi(x)psi(x). Abstract quantum mechanics represents the same physical state as a vector in Hilbert space, written

angle$$ The wavefunction is one representation of this abstract vector: $$psi(x)=langle x|psi angle$$ Here $|x angle$ represents a position basis state. The abstract state contains the physical information; the wavefunction is its coordinate description in the position basis. ## Hilbert space A Hilbert space is a complex vector space with an inner product and a completeness property. For quantum mechanics, this means we can add states, multiply them by complex numbers, compute overlaps, and take limits of state sequences in a well-behaved way. The inner product between two states is written $$langle phi|psi angle$$ The norm of a state is $$||psi||=sqrt{langlepsi|psi angle}$$ Physical states are usually normalized: $$langlepsi|psi angle=1$$ ## Superposition If $|psi_1 angle$ and $|psi_2 angle$ are allowed states, then $$|psi angle=a|psi_1 angle+b|psi_2 angle$$ is also an allowed state, where $a$ and $b$ are complex numbers. This superposition principle is responsible for interference, spin combinations, energy expansions, and quantum computation. The complex coefficients are probability amplitudes when the basis states represent possible measurement outcomes. ## Bases A basis is a set of states that can be used to expand other states. In a finite-dimensional system, an orthonormal basis ${|n angle}$ satisfies $$langle m|n angle=delta_{mn}$$ and a state can be written $$|psi angle=sum_n c_n|n angle$$ where $$c_n=langle n|psi angle$$ If the state is normalized, then $$sum_n |c_n|^2=1$$ ## Continuous bases Position states form a continuous basis. Instead of sums, we use integrals: $$|psi angle=int psi(x)|x angle dx$$ The position basis satisfies a delta-function normalization: $$langle x|x' angle=delta(x-x')$$ This notation unifies discrete and continuous quantum systems. ## Global phase Multiplying a state by an overall phase does not change physical predictions: $$|psi angle ightarrow e^{ialpha}|psi angle$$ All probabilities and expectation values remain the same. Relative phases between components, however, are physically important because they affect interference. ## The big idea Hilbert space is the abstract arena of quantum states. Wavefunctions are representations of state vectors in a chosen basis. Superposition, inner products, normalization, basis expansion, and complex amplitudes form the mathematical structure behind quantum measurement and dynamics.

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