This brief article in the first section brings out the fact that the Heisenberg and the Schrodinger operators become identical if the operators correspond to a conserved quantity. In Section II solutions we show that for time independent operators 𝐴𝑆 that satisfy [𝐴𝑆,𝐻]=0 where 𝐻 satisfies the Schrodinger equation 𝑖ℏ𝜕𝜓𝜕𝑡=𝐻𝜓 we have 𝐴𝑆=𝑎 where ‘a’ is an eigen value of the stated equation. It is independent of space and time coordinates.
This brief article in the first section brings out the fact that the Heisenberg and the Schrodinger operators become identical if the operators correspond to a conserved quantity. In Section II solutions we show that for time independent operators 𝐴𝑆 that satisfy [𝐴𝑆,𝐻]=0 where 𝐻 satisfies the Schrodinger equation 𝑖ℏ𝜕𝜓𝜕𝑡=𝐻𝜓 we have 𝐴𝑆=𝑎 where ‘a’ is an eigen value of the stated equation. It is independent of space and time coordinates.
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
Abstract A basic postulate of wave mechanics is that the wave function of a microscopic system develops in time according to the equation iℏ∂ψ/∂t = Hψ, where H, the Hamiltonian, is an operator which in general depends upon the time. If, and only if, the Hamiltonian is time-independent, then the solutions of this equation take the form ψ(q, t) = ∑ncnѰn(q)e-1Ent/ℏ, (2) where the individual terms Ѱn(q) are functions of the co-ordinates alone and the En are the corresponding eigenvalues of the Hamiltonian, satisfying HѰn = EnѰn. (3)
The time-independent Schrodinger equation: En= -1/2m [d/dx dWn/dx] / Wn(x) + V(x) ((1)) matches a classical conservation of energy equation where -1/2m [d/dx dWn/dx] / Wn(x) = KEclassical(x). In classical physics, however, one has x(t) so each x point is associated with a specific t for a given energy En. In the quantum time-independent equation, one has exp(-iEnt) as the time dependent factor and does not consider any link with x although if one is comparing ((1)) to a classical conservation equation then a link between x and t could be made to complete the analogy. This is usually not considered, but we examine this point as we argue it becomes important in the time-dependent situation. For the time-dependent Schrodinger case we consider two approaches. First, a linear combination of time independent solutions is taken i.e. W(x,t) = Sum over n an(En) exp(-iEnt) Wn(x) ((2)) Again each Wn(x) has an association with KEclassical(x) and so one may think in terms of (x,t) classical pairs for a given En. The second approach is to consider the function: exp(-iEnt + m Integral (0,x) v(y) dy) ((3)). This satisfies the time-dependent Schrodinger equation and yields a classical conservation of energy equation with v(x) being the classical velocity. Noting that En= .5m v(x)v(x) + V(x), ((3)) may be rewritten as: exp(i Integral L dt) where L = T-V is the Langrangian with T=.5mv(x)v(x). At this point En has been removed from the picture, but is replaced with: x(t)= X g(t)/g(T) ((4)) where (X,T) are a space-time pair corresponding to a particular energy E value. Using exp(i Action) ((5)) in the time-dependent Schrodinger equation yields a conservation equation which holds for a particular classical energy as long as (x,t) values in the equation are classically linked to the appropriate classical (X,T) values i.e. for a classical trajectory there is a specific family of X at time T for a given energy. For another energy, there is another family of (X,T) pairs. Thus the equation obtained represents energy conservation for many different energy values depending on which simultaneous values one uses for x and t. There is, however, no overlap as a specific (X,T) family corresponds to an energy E and another nonoverlapping family of (X,T) to another energy. This, however, is the same situation as ((2)) for which there are also families of (X,T) corresponding to each En. The only difference is that the En are quantized i.e. discrete. ((3)) and ((5)) are both solutions to the time-dependent Schrodinger equation which represent a variety of energies and families of (X,T) associated to the classical solution of the energy in question. We thus suggest that ((2)) and ((4)) are equivalent for this reason without using the idea of a complete set of basis functions represented by Wn(x). We consider these ideas as the Schrodinger propagator is given by ((5)), but with ((4)) generalized to: x(t)= Xi g(tf-t)/g(tf-ti) + Xf g(t-ti)/g(tf-ti) for the case that g(0)=0 (2) so that Xi,Xf, tf and ti enter the picture. This then should be equivalent to: Sum over n exp(-iEntf) Wn(xf) Wn* (xi) ((6)) In this note, we suggest that one may link ((5)) and ((6)) based on ideas of space-time families associated with different energies as well as the two being solutions of the time-dependent Schrodinger equation.
The Hamiltonian for a system of n particles that are identical, that is, particles that can be substituted for each other with no physical change, must be completely symmetric under any interchange of its arguments. Let Ψ(1, 2, …, n) be any solution of the Schrödinger equation, depending on the coordinates (spatial and spin) of the n identical particles 1 to n, and let P be any permutation of the n numbers 1 to n. Then PΨ(1, 2, …, n) ≡ Ψ(P1, P2, …, Pn) is a function which depends on the coordinates of particle Ψ in the same manner as the original function Ψ depended on the coordinates of particle i. Then using the symmetry of H, we can show that the operator P commutes with the Hamiltonian ( 2-1 ) H ( P Ψ ) = P ( H Ψ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq89.tif"/>
The solutions to the Time‐Independent Schrödinger Equation (TISE) can be found as long as we can define the Hamiltonian, which in the simplest cases, means defining the potential in which the particle of interest is moving. In the following sections, we will consider a single particle of mass, m p , moving in a potential that is time‐independent, but can vary in space. We will follow four steps: Define the Hamiltonian by defining the Potential Energy Operator since H ^ = - ħ 2 2 m p ∇ ^ 2 + V ^ ( r ) $ \hat{H} = - \frac{{{\hbar}^{2} }}{{2m_{p} }}\hat{\nabla }^{2} + \hat{V}(r) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_1.tif"/> . Solve the TISE to get the wave functions ψ ( x , y , z ) $ \psi (x, y, z) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_2.tif"/> and then the total stationary state wave functions Ψ ( x , y , z ; t ) = ψ ( x , y , z ) e - i ω t . $ \iPsi (x, y, z;t) = \psi (x, y, z)e^{ - i\omega t} . $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_3.tif"/> Determine the energies of the various states of the system. Examine the implications of the solutions to the Schrödinger equations.