Wave function

Citation from wikipedia:

For now, consider the simple case of a non-relativistic single particle, without spin, in one spatial dimension. More general cases are discussed below.

Position-space wave functionsEdit

The state of such a particle is completely described by its wave function,

${\displaystyle \Psi (x,t)\,,}$

where x is position and t is time. This is acomplex-valued function of two real variablesx and t.

For one spinless particle in 1d, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number

${\displaystyle \left|\Psi (x,t)\right|^{2}={\Psi (x,t)}^{*}\Psi (x,t)=\rho (x,t),}$

is interpreted as the probability density that the particle is at x. The asterisk indicates thecomplex conjugate. If the particle's position ismeasured, its location cannot be determined from the wave function, but is described by aprobability distribution. The probability that its position x will be in the interval a ≤ x ≤ b is the integral of the density over this interval:

${\displaystyle P_{a\leq x\leq b}(t)=\int \limits _{a}^{b}dx\,|\Psi (x,t)|^{2}}$

where t is the time at which the particle was measured. This leads to the normalization condition:

${\displaystyle \int \limits _{-\infty }^{\infty }dx\,|\Psi (x,t)|^{2}=1\,,}$

because if the particle is measured, there is 100% probability that it will be somewhere.

For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). Technically, because of the normalization condition, wave functions form a projective space rather than an ordinary vector space. This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of two wave functions Ψ₁ and Ψ₂ can be defined as the complex number (at time t)^{[nb 1]}

${\displaystyle (\Psi _{1},\Psi _{2})=\int \limits _{-\infty }^{\infty }dx\,\Psi _{1}^{*}(x,t)\Psi _{2}(x,t).}$

More details are given below. Although the inner product of two wave functions is a complex number, the inner product of a wave function Ψ with itself,

${\displaystyle (\Psi ,\Psi )=\|\Psi \|^{2}\,,}$

is always a positive real number. The number||Ψ|| (not ||Ψ||²) is called the norm of the wave function Ψ, and is not the same as themodulus |Ψ|.

If (Ψ, Ψ) = 1, then Ψ is normalized. If Ψ is not normalized, then dividing by its norm gives the normalized function Ψ/||Ψ||. Two wave functions Ψ₁ and Ψ₂ are orthogonal if(Ψ₁, Ψ₂) = 0. If they are normalized andorthogonal, they are orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions. In a linear combination of orthogonal wave functions Ψ_nwe have,

${\displaystyle \Psi =\sum _{n}a_{n}\Psi _{n}\,,\quad a_{n}={\frac {(\Psi _{n},\Psi )}{(\Psi _{n},\Psi _{n})}}}$

If the wave functions Ψ_n were nonorthogonal, the coefficients would be less simple to obtain.

In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number

${\displaystyle \left|(\Psi _{1},\Psi _{2})\right|^{2}=P\left(\Psi _{2}\rightarrow \Psi _{1}\right)\,,}$

which, assuming both wave functions are normalized, is interpreted as the probability of the wave function Ψ₂ "collapsing" to the new wave function Ψ₁ upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with Ψ₁being an eigenvector of the resulting eigenvalue. This is the Born rule,^[8] and is one of the fundamental postulates of quantum mechanics.

At a particular instant of time, all values of the wave function Ψ(x, t) are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In Bra–ket notation, this vector is written

${\displaystyle |\Psi (t)\rangle =\int dx\Psi (x,t)|x\rangle }$

and is referred to as a "quantum state vector", or simply "quantum state".There are several advantages to understanding wave functions as representing elements of an abstract vector space:

• All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example:
  • Linear algebra explains how a vector space can be given a basis, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.
  • Bra–ket notation can be used to manipulate wave functions.
• The idea that quantum states are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.

The time parameter is often suppressed, and will be in the following. The x coordinate is a continuous index. The |x⟩ are the basis vectors, which are orthonormal so their inner product is a delta function;

${\displaystyle \langle x'|x\rangle =\delta (x'-x)}$

thus

${\displaystyle \langle x'|\Psi \rangle =\int dx\Psi (x)\langle x'|x\rangle =\Psi (x')}$

and

${\displaystyle |\Psi \rangle =\int dx|x\rangle \langle x|\Psi \rangle =\left(\int dx|x\rangle \langle x|\right)|\Psi \rangle }$

which illuminates the identity operator

${\displaystyle I=\int dx|x\rangle \langle x|\,.}$

Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis).