Many-worlds interpretation

Many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse.

Many-worlds interpretation says there are lots of worlds in reality. Here I will explain what many worlds interpretation is.

For example, suppose you are observing one electron which is inside a small box. If the box is as small as a sugar cube, this electron can be regarded as it is following the classical physics. But if the box is really small - almost as small as the electron's "orbit" or "orbital" - it is very difficult to determine where the electron is precisely.

Actually it's impossible to determine where the electron is in the orbit. Location of the electron is not determined until you open the small box and check the location of the electron (Only probability of where the electron is can be expected by calculation). When I say "the location is not determined," it is not really determined -  The electron exists just as "cloud of probability." [1][2] Thus electron's "orbit" is not really orbit anymore - this is the reason why it's called "orbital" in quantum physics.[3]

We can't expect where the electron really is before we open the box and check where the electron is. This is not because the theory is imperfect, on the contrary, the electron is really cloud of probability before being checked the state. The state of electron is described by wave function.

In many worlds interpretation, this wave function can be applied to the observer  (although in Copenhagen interpretation, you can't apply it to the observer). Our body consists of a lot of atoms and electrons which follow the physical rules. If we apply wave function to the body of the observer, state of the observer and the electron which is inside the small box should be uniquely determined.

If we apply wave function to the observer and the electron being inside the box, we will find that many states are existing simultaneously and every time we observe an object, branches of the state are spawned.

We suppose you would decide to go home from the laboratory if you find the electron at right side of the box, but decide to stay at the lab and keep studying there if you find the electron at left side of the box. Both of the states (or worlds) exist in reality. But you can perceive only one state out of the many states.


[1] Purdue university, Chemical Education Division Groups, http://chemed.chem.purdue.edu/genchem/history/schrodinger.html.
[2] "The life and death of stars", http://www.weinberg.northwestern.edu/web/faculty/, Weinberg college of arts and science.
[3] "What is the difference between an orbit and an orbital?", 2016, Quora, https://www.quora.com/What-is-the-difference-between-an-orbit-and-an-orbital

Reference

[1] Toshifumi Sakaguchi, http://quantum.la.coocan.jp/tutorial.html

[2] Toshifumi Sakaguchi, http://quantum.la.coocan.jp/index_eng.html

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).