page_title: Measurement in quantum mechanics
section_title: Examples
hierarchical_section_title: Measurement in quantum mechanics / Mathematical formalism / Examples
caption_attribution_description: Deutsch: Quantenmechanischer Harmonischer Oszillator einer Dimension
context_page_description: In quantum physics, a measurement is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.
  Quantum physics has proven to be an empirical success and to have wide-ranging applicability. However, on a more philosophical level, debates continue about the meaning of the measurement concept.
context_section_description: The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. A pure state for a qubit can be written as a linear combination of two orthogonal basis states  and  with complex coefficients:
  A measurement in the  basis will yield outcome  with probability  and outcome   with probability , so by normalization,
  An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for  self-adjoint matrices:
  where the real numbers  are the coordinates of a point within the unit ball and
  POVM elements can be represented likewise, though the trace of a POVM element is not fixed to equal 1. The Pauli matrices are traceless and orthogonal to one another with respect to the Hilbert–Schmidt inner product, and so the coordinates  of the state  are the expectation values of the three von Neumann measurements defined by the Pauli matrices. If such a measurement is applied to a qubit, then by the Lüders rule, the state will update to the eigenvector of that Pauli matrix corresponding to the measurement outcome. The eigenvectors of  are the basis states  and , and a measurement of  is often called a measurement in the "computational basis." After a measurement in the computational basis, the outcome of a  or  measurement is maximally uncertain.
  A pair of qubits together form a system whose Hilbert space is 4-dimensional. One significant von Neumann measurement on this system is that defined by the Bell basis, a set of four maximally entangled states:
  A common and useful example of quantum mechanics applied to a continuous degree of freedom is the quantum harmonic oscillator. This system is defined by the Hamiltonian
  where , the momentum operator  and the position operator  are self-adjoint operators on the Hilbert space of square-integrable functions on the real line. The energy eigenstates solve the time-independent Schrödinger equation:
  These eigenvalues can be shown to be given by
  and these values give the possible numerical outcomes of an energy measurement upon the oscillator. The set of possible outcomes of a position measurement on a harmonic oscillator is continuous, and so predictions are stated in terms of a probability density function  that gives the probability of the measurement outcome lying in the infinitesimal interval from  to .
caption: Probability density 
    
      
        
          
            P
            
              n
            
          
          (
          x
          )
        
      
      {\displaystyle P_{n}(x)}
    
   for the outcome of a position measurement given the energy eigenstate 
    
      
        
          
            |
          
          n
          ⟩
        
      
      {\displaystyle |n\rangle }
    
   of a 1D harmonic oscillator.
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