Schrödinger Equation!


In quantum physics, state functions Ψ satisfy equations of the form AΨ = aΨ. Here A is an operator corresponding to some physical quantity, such as energy or momentum, and we want to solve this equation to find a state Ψ that has a specific real value a for the physical quantity represented by the operator. The value a is called the eigenvalue, and the solution Ψ is called an eigenstate of A. To make that clear, the specific solution is usually written as Ψa. The Schrödinger equation results from asking, what is the operator H that corresponds to the (non-relativistic) total energy of a system?  This amounts to asking, what are the operators for the kinetic and potential energies of a system?  Such operators can be specified only in specific coordinate systems, the most common being (r,t) and (x,t). Schrödinger found that the operator for total energy H is given simply by H = iℏ∂/∂t. Solving HΨ(t) = EΨ(t) we instantly find that Ψ(t) = C exp(−iEt/ℏ), where C is an arbitrary constant.  If we work in terms of eigenstates of E, then we can solve instead an equation of the form (K + V)Ψ = EΨ and we now need operators for the kinetic (K) and potential (V) energies of the system. Schrödinger found that the momentum is represented by the operator p = −iℏ∇, or in one dimension, just px = −iℏ∂/∂x. Then the kinetic energy operator for a non-relativistic system is K = p2/(2m) or −ℏ2∇2/(2m).  In one dimension, of course, then K = −((ℏ2)/(2m))(∂2/∂x2).  [A reminder of what the ∇ operator is.]





Notice that the solutions to this equation are real.

For a particle of definite energy, we can always write Ψ(r,t) = ψ(r)exp(−iEt/ℏ) and of course we can make the same separation for Ψ(x,t). Therefore given a potential that is only a function of r or of x, we have a relatively simple equation to solve for ψ. The number of quantum numbers needed to specify the eigenstates of a system depends upon the number of space degrees of freedom the system has.  A system with only one degree of freedom (for example, it can only be found somewhere on the x axis) needs only one quantum number to label the energy eigenvalues.

Vitally important facts about these general relationships:

  • • The probability distribution for a state of definite energy is independent of time! This follows directly from the energy-time uncertainty relation. An energy eigenstate is for that reason often called a “stationary state.”

  • • The state functions ψ(r) or ψ(x) can be real; the differential equations for them may not contain i.

  • • For a free particle, V(r) = 0, or V(x) = 0, the full Schrödinger equation can be solved almost by inspection. The result is, for example, Ψ(r,t) = C exp[(i/ℏ)(p·r - Et)], or ψ(x,t) = C exp[(i/ℏ)(px - Et)]. These expressions are frequently written in terms of the wave number defined by p = ℏk and the angular frequency defined by E = ℏω. C is an arbitrary constant.  The free particle solution is an eigenstate of momentum, so it is defined over all space, and P as a function of r or x is a constant everywhere, as required by the uncertainty relation.

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    A very simple example that incorporates many features of quantum physics is a particle with one degree of freedom, confined between two impenetrable barriers a distance L apart.  This gives us a specific boundary condition, ψ = 0 when x corresponds to either boundary wall. The equation for ψ is then easily solved. It is no accident that the probability amplitudes resemble classical standing waves, since these have equivalent boundary conditions,  no transverse motion at the fixed ends of the string! The standing probability waves are a result of the superposition of an amplitude moving to the right and one moving to the left, since the state cannot possess information about the direction of p. In any 1-dimensional example, energies are labelled by a single so-called principal quantum number n. The lowest energy level may have n = 1, and n increases in steps of 1. Loosely speaking, n is like the number of half-wavelengths inside the barriers.  We can see for the n = 1 state that λ = 2L, so that p = h/2L for that state, the so-called ground state, and thus E1 = (h2/(8mL2)).

    STANDING WAVES ON A ROPE


    Remember a nanometer is 10−9 meters.

    In a finite potential well, the state functions can have nonzero values outside the potential region, but the probability amplitudes there must fall off exponentially, because since V is larger than E there,  the momentum there is imaginary!   The wave number k is proportional to the square root of E - V, so if V becomes larger than E, the wave number k becomes iκ where κ is real, and so exp(ikx) becomes exp(-κx). Because the effective wavelengths are thus longer, the momenta and total energies are smaller, compared to the infinite potential well case.  The finite well only contains a few bound states, whereas the infinite well contains an infinite number. An important idea in all cases of bound states is the parity of a given solution. A solution that satisfies ψ(x) = ψ(−x) is called a positive parity solution, while one that satisfies ψ(x) = −ψ(−x) is called a negative parity solution.  All state functions generally are eigenstates of parity.  However, it is remarkable that parity turns out not to be a conserved quantity for weak interactions!

    Consider a 1-D harmonic oscillator potential, V(x) = (1/2)kx2. Using the usual HO relation between k, m and ω, we can write it as V(x) =(1/2)mω2x2. The levels predicted by solving the equation turn out to be equally spaced by ℏω, and since the potential is infinite, there are an infinite number of levels.  The lowest energy level has energy (1/2)ℏω. En = ℏω(n + (1/2)). The principal quantum number n ranges from 0 to infinity.  Note that of course quantum harmonic oscillator solutions do not oscillate!  For the states, P(x) has no time dependence at all.  We call it a quantum harmonic oscillator because its potential energy has the same form as the classical oscillator.   The quantum harmonic oscillator is one of the most useful systems in quantum physics, because so many physical systems can be modeled as similar to such oscillators.  This is because any potential energy which has  a local minimum will have oscillator-like solutions for small excitations in the minimum!  Even fundamental quantum fields of force can be modeled in terms of relativistic harmonic oscillators!


    Probability distributions of the oscillator states

    Bound states illustrate two basic features of quantum physics. First, the bound state functions we have discussed can be normalized, by which we mean that we can choose an overall constant such that ∫ψ(x)2dx = 1. When this is done, the probability distribution P(x) will give the absolute, rather than relative, probability of finding the particle at x. The other important property that can be easily illustrated is that state functions belonging to different quantum numbers are orthogonal, that is, ∫ψn(x)ψm(x)dx = 0 when n ≠ m. The integrals can be taken to extend from minus infinity to plus infinity.   With normalized state functions, you can calculate the expectation value of any operator A, using ∫Ψ*AΨ dx.


    Because the Laplacian operator ∇2 in cartesian coordinate form does not mix x, y and z coordinates, it is possible to do a very simple 3-dimensional example using our previous results. Imagine a quantum particle inside an impenetrable cube of length L on a side. The Schrödinger equation can be separated into three different equations, one for each of the coordinates x, y and z, with the full state function given just as a product of the three independent solutions. This is called the separation of variables method. In other words we would have something like ψ(x,y,z) = X(x)Y(y)Z(z) where each term in the product is the infinite 1D well solution. This approach is generally used to tackle ANY 3D problem in quantum physics.

    This example allows us to demonstrate one of the most important features of quantum physics. When two or more different states have the same energy, this situation is called degeneracy. In quantum physics, every degeneracy corresponds to a symmetry of the system. A symmetry is a change that can be made in some aspect of a system, that has no physical consequences, in other words, leaves the physics of the system unaffected. In the case of the cubical box, the symmetry involves relabeling the axes x, y and z. Clearly the choice of which axis is called x, which is called y and which is called z is completely arbitrary. The result is that for the set of quantum numbers (nx,ny,nz), the state (1,2,3) has precisely the same energy as the state (2,3,1) or the state (3,1,2).


    In convenient units, ℏ is about 200 × 10−24 eV-seconds.  Other units are tabulated here. A convenient unit for atomic and molecular distances is a0, the so-called Bohr radius, which is about 5.3 x 10−11 meters.




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