Quantum Mechanics for Scientists and Engineers David Miller
The particle in a box
The particle in a box Linearity and normalization
Linearity and Schrödinger s equation We see that Schrödinger s equation is linear m V r E The wavefunction appears only in first order there are no second or higher order terms such as or 3 So, if is a solution, so also is a this just corresponds to multiplying both sides by the constant a
Normalization of the wavefunction Born postulated the probability Pr of finding a particle near a point r is r Specifically let us define P r as a probability density For some very small (infinitesimal) volume d 3 r around r the probability of finding the particle in that volume is P r d r 3
Normalization of the wavefunction The sum of all such probabilities should be 1 So 3 Prd r 1 r Can we choose so that we can use r as the probability density not just proportional to probability density? Unless we have been lucky our solution to Schrödinger s equation did not give a r so that 3 d r r 1
Normalization of the wavefunction Generally, this integral would give some other real positive number which we could write as 1/ a where a is some (possibly complex) number That is, 3 1 r d r a But we know that if r is a solution of Schrödinger s equation so also is a r
Normalization of the wavefunction So if we use the solution N a instead of then 3 N r d r 1 and we can use N r as the probability density, i.e., N r r P N r would then be called a normalized wavefunction
Normalization of the wavefunction So, to summarize normalization we take the solution we have obtained from Schrödinger s wave equation we integrate r to get a number we call 1/ a then we obtain the normalized wavefunction a for which N 3 N r d r 1 and we can use N r as the probability density
Technical notes on normalization Note that normalization only sets the magnitude of a not the phase we are free to choose any phase for a or indeed for the original solution a phase factor expi is just another number by which we can multiply the solution and still have a solution
Technical notes on normalization If we think of space as infinite functions like sin kx,, and exp ikr cannot be normalized in this way Technically, their squared modulus is not Lebesgue integrable They are not L functions This difficulty is mathematical, not physical It is caused by over-idealizing the mathematics to get functions that are simple to use cos kz
Technical notes on normalization There are work-arounds for this difficulty 1 - only work with finite volumes in actual problems this is the most common solution - use normalization to a delta function introduces another infinity to compensate for the first one This can be done but we will try to avoid it
The particle in a box Solving for the particle in a box
Particle in a box We consider a particle of mass m with a spatially-varying potential V(z) in the z direction so we have a Schrödinger equation z d V z z E z m dz where E is the energy of the particle and (z) is the wavefunction
Particle in a box Suppose the potential energy is a simple rectangular potential well thickness L z Potential energy is constant inside we choose V 0 there rising to infinity at the walls i.e., at z 0 and z Lz We will sometimes call this an infinite or infinitely deep (potential) well Energy V 0 z 0 L z z L z
Particle in a box Because these potentials at z 0 and at z L z are infinitely high but the particle s energy E is presumably finite we presume there is no possibility of finding the particle outside i.e., for z 0 or z Lz so the wavefunction is 0 there so should be 0 at the walls Energy V 0 z 0 L z z L z
Particle in a box With these choices inside the well the Schrödinger equation d z V z z E z m dz becomes d z E z m dz with the boundary conditions and 0 0 L z 0 Energy V 0 z 0 L z z L z
Particle in a box The general solution to the equation d z E z m dz is of the form z Asin kz Bcos kz where A and B are constants and k me/ The boundary condition 0 0 means B 0 because cos 0 1 Energy V 0 z 0 L z z L z
Particle in a box With now z Asin kz and the condition L z 0 kz must be a multiple of, i.e., k me/ n / Lz where n is an integer k Since, therefore, E m the solutions are n z n nz Ansin with En L z m Lz Energy n 3 n n 1 E 3 E E 1
Particle in a box We restrict n to positive integers n 1,, for the following reasons Since sinasina for any real number a the wavefunctions with negative n are the same as those with positive n within an arbitrary factor, here -1 the wavefunction for n 0 is trivial the wavefunction is 0 everywhere Energy n n 3 n n 1 z E 3 E E 1 n z Ansin Lz
Particle in a box We can normalize the wavefunctions L z n z L z An sin dz An 0 Lz To have this integral equal 1 choose An / Lz Note A n can be complex All such solutions are arbitrary within a unit complex factor Energy n 3 n n 1 E 3 E E 1 Conventionally, we choose A n real for simplicity in writing
Particle in a box E n n m Lz n 3 E 3 n z n z sin Lz Lz Energy n E n 1,, 0 n 1 E 1 L z
The particle in a box Nature of particle-in-a-box solutions
Eigenvalues and eigenfunctions Solutions with a specific set of allowed values of a parameter (here energy) eigenvalues and with a particular function associated with each such value eigenfunctions can be called eigensolutions n E n z n m Lz n z sin Lz Lz n 1,,
Eigenvalues and eigenfunctions Here since the parameter is an energy we can call the eigenvalues eigenenergies and we can refer to the eigenfunctions as the energy eigenfunctions n E n z n m Lz n z sin Lz Lz n 1,,
Degeneracy Note in some problems it can be possible to have more than one eigenfunction with a given eigenvalue a phenomenon known as degeneracy The number of such states with the same eigenvalue is called the degeneracy of that state
Parity of wavefunctions Note these eigenfunctions have definite symmetry n 3 the n 1 function is the mirror image on the left of what it is on the right n such a function has even parity or is said to be an even function The n 3 eigenfunction is also even n 1
Parity of wavefunctions The n eigenfunction is an inverted image the value at any point on the right of the center is exactly minus the value at the mirror image point on the left of the center Such a function has odd parity or is said to be an odd function n 3 n n 1
Parity of wavefunctions For this symmetric well problem the functions alternate between being even and odd and all the solutions are either even or odd i.e., all the solutions have a definite parity Such definite parity is common in symmetric problems it is mathematically very helpful n 3 n n 1
Quantum confinement This particle-in-a-box behavior is very different from the classical case 1 there is only a discrete set of possible values for the energy there is a minimum possible energy for the particle corresponding to n 1 here E 1 / m / Lz sometimes called a zero-point energy Energy 0 n 3 n n 1 L z E 3 E E 1
Quantum confinement 3 - the particle is not uniformly distributed over the box, and its distribution is different for n 3 different energies It is almost never found very near to the walls of the box n the probability obeys a standing wave pattern 0 n 1 Energy E 3 E E 1 L z
Quantum confinement In the lowest state ( n 1 ), it is most likely to be found near the center of the box In higher states, n 3 E 3 there are points inside the box where the particle will never be found Energy 0 n n 1 E E 1 L z
Quantum confinement Note that each successively higher energy state has one more zero in the eigenfunction This is very common behavior in quantum mechanics Energy 0 n 3 n n 1 E 3 E E 1 L z
Energies in quantum mechanics In quantum mechanical calculations we can always use Joules as units of energy but these are rather large A very convenient energy unit which also has a simple physical significance 19 is the electron-volt (ev) 1.6010 J the energy change of an electron in moving through an electrostatic potential change of 1V Energy in ev = energy in Joules/e 19 e electronic charge 1.60176 56510 C (Coulombs)
Orders of magnitude E.g., confine an electron in a 5 Å (0.5 nm) thick box The first allowed level for the electron is o E m 10 19 1 / / 510.410 J1.5 ev The separation between the first and second allowed energies ( E ) E1 3E1 is 4.5eV which is a characteristic size of major energy separations between levels in an atom