kb/data/en.wikipedia.org/wiki/Bra–ket_notation-6.md

---
title: "Bra–ket notation"
chunk: 7/8
source: "https://en.wikipedia.org/wiki/Bra–ket_notation"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T14:40:03.882193+00:00"
instance: "kb-cron"
---

  
    
      
        
          |
        
        ψ
        ⟩
        
          |
        
        ϕ
        ⟩
        
        ,
        
        
          |
        
        ψ
        ⟩
        ⊗
        
          |
        
        ϕ
        ⟩
        
        ,
        
        
          |
        
        ψ
        ϕ
        ⟩
        
        ,
        
        
          |
        
        ψ
        ,
        ϕ
        ⟩
        
        .
      
    
    {\displaystyle |\psi \rangle |\phi \rangle \,,\quad |\psi \rangle \otimes |\phi \rangle \,,\quad |\psi \phi \rangle \,,\quad |\psi ,\phi \rangle \,.}
  

See quantum entanglement and the EPR paradox for applications of this product.

== The unit operator ==
Consider a complete orthonormal system (basis),

  
    
      
        {
        
          e
          
            i
          
        
         
        
          |
        
         
        i
        ∈
        
          N
        
        }
        
        ,
      
    
    {\displaystyle \{e_{i}\ |\ i\in \mathbb {N} \}\,,}
  

for a Hilbert space H, with respect to the norm from an inner product ⟨·,·⟩.
From basic functional analysis, it is known that any ket 
  
    
      
        
          |
        
        ψ
        ⟩
      
    
    {\displaystyle |\psi \rangle }
  
 can also be written as

  
    
      
        
          |
        
        ψ
        ⟩
        =
        
          ∑
          
            i
            ∈
            
              N
            
          
        
        ⟨
        
          e
          
            i
          
        
        
          |
        
        ψ
        ⟩
        
          |
        
        
          e
          
            i
          
        
        ⟩
        ,
      
    
    {\displaystyle |\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}|\psi \rangle |e_{i}\rangle ,}
  

with ⟨·|·⟩ the inner product on the Hilbert space.
From the commutativity of kets with (complex) scalars, it follows that

  
    
      
        
          ∑
          
            i
            ∈
            
              N
            
          
        
        
          |
        
        
          e
          
            i
          
        
        ⟩
        ⟨
        
          e
          
            i
          
        
        
          |
        
        =
        
          I
        
      
    
    {\displaystyle \sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|=\mathbb {I} }
  

must be the identity operator, which sends each vector to itself.
This, then, can be inserted in any expression without affecting its value; for example

  
    
      
        
          
            
              
                ⟨
                v
                
                  |
                
                w
                ⟩
              
              
                
                =
                ⟨
                v
                
                  |
                
                
                  (
                  
                    
                      ∑
                      
                        i
                        ∈
                        
                          N
                        
                      
                    
                    
                      |
                    
                    
                      e
                      
                        i
                      
                    
                    ⟩
                    ⟨
                    
                      e
                      
                        i
                      
                    
                    
                      |
                    
                  
                  )
                
                
                  |
                
                w
                ⟩
              
            
            
              
              
                
                =
                ⟨
                v
                
                  |
                
                
                  (
                  
                    
                      ∑
                      
                        i
                        ∈
                        
                          N
                        
                      
                    
                    
                      |
                    
                    
                      e
                      
                        i
                      
                    
                    ⟩
                    ⟨
                    
                      e
                      
                        i
                      
                    
                    
                      |
                    
                  
                  )
                
                
                  (
                  
                    
                      ∑
                      
                        j
                        ∈
                        
                          N
                        
                      
                    
                    
                      |
                    
                    
                      e
                      
                        j
                      
                    
                    ⟩
                    ⟨
                    
                      e
                      
                        j
                      
                    
                    
                      |
                    
                  
                  )
                
                
                  |
                
                w
                ⟩
              
            
            
              
              
                
                =
                ⟨
                v
                
                  |
                
                
                  e
                  
                    i
                  
                
                ⟩
                ⟨
                
                  e
                  
                    i
                  
                
                
                  |
                
                
                  e
                  
                    j
                  
                
                ⟩
                ⟨
                
                  e
                  
                    j
                  
                
                
                  |
                
                w
                ⟩
                
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\langle v|w\rangle &=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)|w\rangle \\&=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)\left(\sum _{j\in \mathbb {N} }|e_{j}\rangle \langle e_{j}|\right)|w\rangle \\&=\langle v|e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|w\rangle \,,\end{aligned}}}
  

where, in the last line, the Einstein summation convention has been used to avoid clutter.
In quantum mechanics, it often occurs that little or no information about the inner product ⟨ψ|φ⟩ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients ⟨ψ|ei⟩ = ⟨ei|ψ⟩* and ⟨ei|φ⟩ of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more.
For more information, see Resolution of the identity,

  
    
      
        
          
            I
          
        
        =
        ∫
        
        d
        x
         
        
          |
        
        x
        ⟩
        ⟨
        x
        
          |
        
        =
        ∫
        
        d
        p
         
        
          |
        
        p
        ⟩
        ⟨
        p
        
          |
        
        ,
      
    
    {\displaystyle {\mathbb {I} }=\int \!dx~|x\rangle \langle x|=\int \!dp~|p\rangle \langle p|,}
  
 where 
  
    
      
        
          |
        
        p
        ⟩
        =
        ∫
        d
        x
        
          
            
              
                e
                
                  i
                  x
                  p
                  
                    /
                  
                  ℏ
                
              
              
                |
              
              x
              ⟩
            
            
              2
              π
              ℏ
            
          
        
        .
      
    
    {\displaystyle |p\rangle =\int dx{\frac {e^{ixp/\hbar }|x\rangle }{\sqrt {2\pi \hbar }}}.}
  

Since ⟨x′|x⟩ = δ(x − x′), plane waves follow, 
  
    
      
        ⟨
        x
        
          |
        
        p
        ⟩
        =
        
          
            
              e
              
                i
                x
                p
                
                  /
                
                ℏ
              
            
            
              2
              π
              ℏ
            
          
        
        .
      
    
    {\displaystyle \langle x|p\rangle ={\frac {e^{ixp/\hbar }}{\sqrt {2\pi \hbar }}}.}
  

In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate 
  
    
      
        
          |
        
        ϖ
        ⟩
        =
        
          lim
          
            p
            →
            0
          
        
        
          |
        
        p
        ⟩
      
    
    {\textstyle |\varpi \rangle =\lim _{p\to 0}|p\rangle }
  
 in the momentum representation, i.e., 
  
    
      
        
          
            
              p
              ^
            
          
        
        
          |
        
        ϖ
        ⟩
        =
        0
      
    
    {\displaystyle {\hat {p}}|\varpi \rangle =0}
  
. Consequently, the corresponding wavefunction is a constant, 
  
    
      
        ⟨
        x
        
          |
        
        ϖ
        ⟩
        
          
            2
            π
            ℏ
          
        
        =
        1
      
    
    {\displaystyle \langle x|\varpi \rangle {\sqrt {2\pi \hbar }}=1}
  
, and 

  
    
      
        
          |
        
        x
        ⟩
        =
        δ
        (
        
          
            
              x
              ^
            
          
        
        −
        x
        )
        
          |
        
        ϖ
        ⟩
        
          
            2
            π
            ℏ
          
        
        ,
      
    
    {\displaystyle |x\rangle =\delta ({\hat {x}}-x)|\varpi \rangle {\sqrt {2\pi \hbar }},}
  
 as well as 
  
    
      
        
          |
        
        p
        ⟩
        =
        exp
        ⁡
        (
        i
        p
        
          
            
              x
              ^
            
          
        
        
          /
        
        ℏ
        )
        
          |
        
        ϖ
        ⟩
        .
      
    
    {\displaystyle |p\rangle =\exp(ip{\hat {x}}/\hbar )|\varpi \rangle .}
  

Typically, when all matrix elements of an operator such as 
  
    
      
        ⟨
        x
        
          |
        
        A
        
          |
        
        y
        ⟩
      
    
    {\displaystyle \langle x|A|y\rangle }
  
 are available, this resolution serves to reconstitute the full operator,

  
    
      
        ∫
        d
        x
        
        d
        y
        
        
          |
        
        x
        ⟩
        ⟨
        x
        
          |
        
        A
        
          |
        
        y
        ⟩
        ⟨
        y
        
          |
        
        =
        A
        
        .
      
    
    {\displaystyle \int dx\,dy\,|x\rangle \langle x|A|y\rangle \langle y|=A\,.}
  

== Notation used by mathematicians ==
The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space).
Let 
  
    
      
        (
        
          
            H
          
        
        ,
        ⟨
        ⋅
        ,
        ⋅
        ⟩
        )
      
    
    {\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )}
  
 be a Hilbert space and h ∈ H a vector in H. What physicists would denote by |h⟩ is the vector itself. That is,

  
    
      
        
          |
        
        h
        ⟩
        ∈
        
          
            H
          
        
        .
      
    
    {\displaystyle |h\rangle \in {\mathcal {H}}.}
  

Let H* be the dual space of H. This is the space of linear functionals on H. The embedding 
  
    
      
        Φ
        :
        
          
            H
          
        
        ↪
        
          
            
              H
            
          
          
            ∗
          
        
      
    
    {\displaystyle \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{*}}
  
 is defined by 
  
    
      
        Φ
        (
        h
        )
        =
        
          φ
          
            h
          
        
      
    
    {\displaystyle \Phi (h)=\varphi _{h}}
  
, where for every h ∈ H the linear functional 
  
    
      
        
          φ
          
            h
          
        
        :
        
          
            H
          
        
        →
        
          C
        
      
    
    {\displaystyle \varphi _{h}:{\mathcal {H}}\to \mathbb {C} }
  
 satisfies for every g ∈ H the functional equation 
  
    
      
        
          φ
          
            h
          
        
        (
        g
        )
        =
        ⟨
        h
        ,
        g
        ⟩
        =
        ⟨
        h
        ∣
        g
        ⟩
      
    
    {\displaystyle \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle }
  
.
Notational confusion arises when identifying φh and g with ⟨h| and |g⟩ respectively. This is because of literal symbolic substitutions. Let 
  
    
      
        
          φ
          
            h
          
        
        =
        H
        =
        ⟨
        h
        ∣
      
    
    {\displaystyle \varphi _{h}=H=\langle h\mid }
  
 and let g = G = |g⟩. This gives

  
    
      
        
          φ
          
            h
          
        
        (
        g
        )
        =
        H
        (
        g
        )
        =
        H
        (
        G
        )
        =
        ⟨
        h
        
          |
        
        (
        G
        )
        =
        ⟨
        h
        
          |
        
        
          
            (
          
        
        
          |
        
        g
        ⟩
        
          
            )
          
        
        
        .
      
    
    {\displaystyle \varphi _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}