With this we find This is
With this we find
This is all that one can say for , however when there is also a negative corticosterone solution which is a bound state. One must have in this case, up to a normalization constantwith K>0.
The delta potential now imposes or and one has as the energy of the bound state.
The enzyme analog Here we have one quantum variable that represents the substrate–product system and another one representing the conformation of the enzyme–substrate complex. The first will be modeled as a particle on a line as above, the other will have a finite number of conformations and so will be a discrete variable taking on the (conventional) values s=1, 2, …, n. The combined wave function will then be Ψ(q, s, t) which for convenience will be represented as a column matrixThe Hilbert space is then and the enzyme can be thought of as a qudit of dimension n. In general the two variables are entangled. In spite of the attention given to entanglement in the quantum information literature, what is more to the point here is the superposition of the conformation states of the enzyme–substrate complex. We assume that for each conformation of the enzyme, it subjects the other variable to a delta function energy barrier with . The Schrödinger equation for the combined system is now:where is the diagonal matrix of the strengths of the delta potentials and M is a n×n Hermitian matrix describing the quantum dynamics of the enzyme–substrate conformations. In this expression the quantity is the interaction Hamiltonian, and is the free Hamiltonian. The basis in which (9) is written is one in which H is (block) diagonal, and will be referred to as the interaction basis. Unless M is diagonal, then Eq. (9) cannot be solved by pure phase solutions on either side of the q=0 barrier, but a unitary transform of it can be. As M is Hermitian, there is a unitary U such that which are the energy eigenvalues of the enzyme alone. Letting now Eq. (9) becomes Eq. (10) is written in a basis in which the free Hamiltonian is (block) diagonal and will be called the free basis. Non-trivial quantum effects can only happen if the interaction and the free basis are misaligned for otherwise the system of equations separates into independent ones. On either side of the barrier the various components of decouple and satisfy a free particle Schrödinger equation. So we can take, assuming stationary waves now, that where the equation that must satisfy is:from which we deduce that for each s we have phase waves of the formwithfixing thus the value of k for each value of the energy. Continuity at q=0 entails A+B=C+D. Call this common value F. The delta function potentials now impose the following condition on the discontinuity of ψ′(q) at q=0. For tunneling solutions we set D=0 so one has C=A+B=F and we impose ∑k A 2=1 to normalize the flux arriving from the left. The transmission rate is then T=∑k F 2. In this case we have: Let and arrange the A, the B and the F into column vectors A, B, and F respectively. We now have . For fixed A this can be solved for B which can then be used to calculate the transmission coefficient. We have and after a little manipulation we have (compare to (5)) For E greater than the largest e one can take k>0, hence real, and seeing that is Hermitian, the inverse matrix in the formula above exists, and we have a well defined scattering solution. Possible solutions for lower values of E need a more detailed analysis.
Partial financial support was provided by FAPERJ, a government agency of the state of Rio de Janeiro, Brazil.
Introduction Soil zymography is a new technique for in situ visualization and quantification of two-dimensional distribution of enzyme activities in soil and rhizosphere studies (Hoang et al., 2016; Ge et al., 2017; Liu et al., 2017; Razavi et al., 2016, 2017; Sanaullah et al., 2016; Spohn et al., 2013; Spohn and Kuzyakov, 2014). During zymography a membrane saturated with an enzyme-specific fluorogenic substrate is placed on the surface of a soil sample. Upon a contact of the substrate with soil enzymes, a fluorescent product (e.g. MUF: 4-methylumbelliferone, or AMC: 7-amido-4-methylcoumarin) is released and its presence on the membrane is then detected under UV light. The fluorescing pattern on the membrane reflects spatial distribution of active enzymes on the soil surface.