Question

What are the mathematical formulations of quantum mechanics?

Answer 1

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, `Psi = Psi(x,t)`, but `hatH` is not time-dependent. Schrodinger postulated that:

1. The wave function `Psi` was time-dependent, but the operators associated with it are time-independent.
2. The wave function `psi` was an electric charge density spread out over allspace.
3. 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)`

![https://upload.wikimedia.org/]()

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

HEISENBERG FORMULATION

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

1. A system is described by some arbitrary ket vector `| n >>`, known as a state vector, independent of time.
2. 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)`.
3. 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 `hatU_t` in position space:

`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 `(del)/(delt) (Theta(t-t_0)) = delta(t-t_0)`,

`(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 `t/N -> dt`, for a single time step (`N = 1`):

`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.