# What are the mathematical formulations of quantum mechanics?

##### 1 Answer

They are:

- Schrodinger formulation (wave mechanics)
- Heisenberg formulation (matrix mechanics)
- Feynman Path Integral formulation

The major differences are:

- Schrodinger formulated time-dependent wave functions and time-independent operators.
- Heisenberg formulated time-independent ket vectors and time-dependent operators.
- Feynman formulated an integration over all quantum paths, which could be described as the convolution of the Green's function
#G(x,x'; t)# response to an impulse with a weight given by the stationary state#psi(x,t_0)# .

For simplicity we do this in one dimension.

**DISCLAIMER:** *LONG ANSWER!*

**SCHRODINGER FORMULATION**

The **time-dependent Schrodinger equation** is:

#iℏ(delPsi)/(delt) = hatHPsi# where

#Psi(x,t) = e^(-iEt//ℏ)psi(x)# is the time-dependent wave function and#hatH# is the time-independent Hamiltonian of the system.

As you can see,

- The wave function
#Psi# was time-dependent, but the operators associated with it are time-independent. - The wave function
#psi# was an electric charge density spread out over allspace. - With the input of Max Born,
#psi# is the probability amplitude, while#int_"allspace" psi^"*"psi d tau# is the probability of finding the electron somewhere.

In essence, Schroedinger was a guy who believed in wave mechanics and particle-wave duality, i.e. that quantum particles could be described by the de Broglie relation:

#lambda = h/(mv)#

This is probably the easiest to understand out of all the formulations.

**HEISENBERG FORMULATION**

Heisenberg, for the life of him, saw it this way:

- A system is described by some arbitrary ket vector
#| n >># , known as a state vector, independent of time. - The inner product of
#| n >># with#<< x |# describing all the possible position bra vectors gives a complete set of eigenstates#<< x | n >> = psi_n(x)# . - The operators could be functions of time, and any time translation operator is unitary.

Under this formulation, the **time-dependent Schrodinger equation** becomes:

#iℏ (d << x | n >>)/(dt) = hatH << x | n >>#

From this, one could formulate the following relationships:

**Overlap of definite position and momentum**

#<< x | p_x >> = 1/sqrt(2piℏ) e^(ip_x x//ℏ)#

**Matrix product of position and momentum with themselves**

#int_(-oo)^(oo)# # | x >> << x |# #dx = int_(-oo)^(oo)# # | p_x >> << p_x |# #dp_x# #-= 1#

**Dirac Delta function**

#int_(-oo)^(oo) << x | p_x >> << p_x | x' >>dp_x = << x | x' >>#

#= 1/(2piℏ) int_(-oo)^(oo) e^(ip_x(x-x')//ℏ)dp_x = delta(x-x')#

#int_(-oo)^(oo) << p_x | x >> << x | p_x' >>dx = << p_x | p_x' >>#

#= 1/(2piℏ) int_(-oo)^(oo) e^(ix(p_x-p_x')//ℏ)dx = delta(p_x-p_x')#

**Relation between wave functions in position/momentum representations**

#phi_n(p_x) = << p_x | n >>#

#= int_(-oo)^(oo) << p_x | x >> << x | n >> dx#

#= 1/sqrt(2piℏ) int_(-oo)^(oo) e^(-ip_x x//ℏ) psi_n(x) dx#

#psi_n(x) = << x | n >>#

#= int_(-oo)^(oo) << x | p_x >> << p_x | n >> dp_x#

#= 1/sqrt(2piℏ) int_(-oo)^(oo) e^(ip_x x//ℏ) phi_n(p_x) dp_x#

**FEYNMAN PATH INTEGRAL FORMULATION**

Probably the hardest one to understand, I think... Feynman wanted to describe the **transition probability amplitude** of the quantum particle by summing over all possible quantum paths:

#| psi (x,t') >># #= int_(-oo)^(oo) << psi(x',t') | psi(x_0, t_0) >> dx' cdot# #| psi(x',t') >>#

Under this formulation, define the unitary time translator

#hatU_t = e^(-ihatH(t-t_0)//ℏ)Theta(t-t_0)# ,where

#Theta(t-t_0) = int_(-oo)^(t) delta(t'-t_0)dt'# is the Heaviside step function and#hatH# is the Hamiltonian of the system.

Then a Green function can be defined in terms of

#G(x,x'; t) = << x | hatU_t | x' >># where

#x'# is the position at which an initial impulse to the system was imparted at time#t_0# , and#x# is the propagated position at time#t# .

The **Feynman Path Integral** is then the Green's function formulation of a system. For instance, if

#iℏ(delPsi)/(del t) = hatH Psi# ,

then since

#(hatH - iℏ(del)/(delt)) G(x,x'; t)#

#= (hatH - iℏ(del)/(delt)) << x | e^(-ihatH(t-t_0)//ℏ)Theta(t-t_0) | x' >>#

#= [ . . . ] = -iℏ delta(x-x')delta(t-t_0)#

As it turns out, if the Hamiltonian of a system is given by, for example,

#hatH = p_x^2/(2m) + V(hatx)# ,

then for a small enough time step

#G(x,x'; t) = [ . . . ]#

#= 1/(2piℏ) int_(-oo)^(oo) e^(-ip_x^2(t-t_0)//2mNℏ)e^(-iV(hatx)(t-t_0)//Nℏ) e^(ip_x(x-x')//ℏ) dp_x#

#= sqrt((Nm)/(2pi iℏ(t-t_0))) "exp"[(i(t-t_0))/(Nℏ) (N^2/2 m((x - x')/(t-t_0))^2 - V(x'))],#

#N = 1#

This does indeed relate back to the wave function:

#Psi_n(x,t) = int_(-oo)^(oo) underbrace(G(x,x'; t))_(<< x | hatU_t | x' >>) underbrace(psi_n(x',t_0))_(<< x' | n >>) dx'#

In a way (and this made the most sense to me):

*The wave function can be given by the integral over the path described by the convolution of the impulse on the system with a weight given by the initial state.*