Time Evolution of Quantum States and Perturbation Theory, Lecture notes of Quantum Mechanics

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Advanced Quantum Mechanics

Judith McGovern

December 20, 2012

This is the web page for Advanced Quantum Mechanics (PHYS30201) for the session 2012/13. The notes are not intended to be fully self-contained, but summarise lecture material and give pointers to textbooks. These notes have been prepared with TEX4ht, and use MathML to render the equations. Try the link to “This course and other resources” to see if your browser is compatible. (In particular there is a problem with square roots of fractions that you should check is OK.) If you are using Internet Explorer, you may need to download “MathPlayer” from here. Please report errors to Judith McGovern.

The advantage of MathML is accessibility (at least in theory), equations that can be mag- nified if you change the magnification of the page (or that pop out with a click using “Math- Player”), and much cleaner directories. The disadvantage is that I don’t think it looks as nice on average, and I don’t know in advance how much trouble might be caused by incompatable browsers. I may revert to using images for equations (as the web notes discussed above do) if there are widespread problems, so let me know.

Operator methods in Quantum

Mechanics

1.1 Postulates of Quantum Mechanics

Summary: All of quantum mechanics follows from a small set of assumptions, which cannot themselves be derived.

There is no unique formulation or even number of postulates, but all formulations I’ve seen have the same basic content. This formulation follows Shankar most closely, though he puts III and IV together. Nothing significant should be read into my separating them (as many other authors do), it just seems easier to explore the consequences bit by bit. I: The state of a particle is given by a vector |ψ(t)〉 in a Hilbert space. The state is normalised: 〈ψ(t)|ψ(t)〉 = 1. This is as opposed to the classical case where the position and momentum can be specified at any given time. This is a pretty abstract statement, but more informally we can say that the wave function ψ(x, t) contains all possible information about the particle. How we extract that information is the subject of subsequent postulates. The really major consequence we get from this postulate is superposition, which is behind most quantum weirdness such as the two-slit experiment. II: There is a Hermitian operator corresponding to each observable property of the particle. Those corresponding to position ˆx and momentum ˆp satisfy [ˆxi, pˆj ] = iℏδij. Other examples of observable properties are energy and angular momentum. The choice of these operators may be guided by classical physics (eg ˆp · ˆp/ 2 m for kinetic energy and ˆx × ˆp for orbital angular momentum), but ultimately is verified by experiment (eg Pauli matrices for spin-^12 particles). The commutation relation for ˆx and ˆp is a formal expression of Heisenberg’s uncertainty principle. III: Measurement of the observable associated with the operator Ω will result in one of thê eigenvalues ωi of Ω. Immediately after the measurement the particle will be in the correspondinĝ eigenstate |ωi〉. This postulate ensures reproducibility of measurements. If the particle was not initially in the state |ωi〉 the result of the measurement was not predictable in advance, but for the result of a measurement to be meaningful the result of a subsequent measurement must be predictable. (“Immediately” reflects the fact that subsequent time evolution of the system will change the value of ω unless it is a constant of the motion.) IV: The probability of obtaining the result ωi in the above measurement (at time t 0 ) is |〈ωi|ψ(t 0 )〉|^2.

• Since |α〉 =

α(x′)|x′〉dx′, from the fourth postulate we might expect that P (x) ≡ |α(x)|^2 is the probability of finding the particle at position x. But we also need

P (x′)dx′^ = 1

• the probability of finding the particle at some x is unity. The transition from sums to integrals corresponds to the transition from discrete probabilities to probability distribu- tions, and so the correct interpretation is that P (x)dx is the probability of finding the particle in an infinitesimal interval between x and x + dx.
• Since 〈x|xˆ|α〉 = x〈x|α〉 = xα(x), we can say that the position operator in the position space representation is simply x, and operating with it is equivalent to multiplying the wave function by x.
• If we make the hypothesis that 〈x|pˆ|α〉 = −iℏdα(x)/dx, we can see that the required commutation relation [ˆx, pˆ] = iℏ is satisfied.
• We can equally imagine states of definite momentum, |p〉, with ˆp|p〉 = p|p〉. Then ˜α(p) = 〈p|α〉 is the wave function in position space; in this representation ˆp is given by p and ˆx by (iℏ)d/dp.
• Let’s write the position-space wave function of |p〉, 〈x|p〉, as φp(x). Since 〈x|pˆ|p〉 can be written as either pφp(x) or −iℏdφp(x)/dx, we see that φp(x) = (1/ √ 2 πℏ) eipx/ℏ. This is just a plane wave, as expected. (The normalisation is determined by 〈p|p′〉 = δ(p − p′).)
• This implies that α(x) = (1/√ 2 πℏ)

eip ′x/ℏ α˜(p′)dp′, that is the position- and momentum- space wave functions are Fourier transforms of each other. (The 1/

ℏ is present in the prefactor because we are using p and not k = p/ℏ as the conjugate variable.)

• Strictly speaking we should write the position-space representations of ˆx and ˆp as 〈x|ˆx|x′〉 = x′δ(x−x′) and 〈x|pˆ|x′〉 = (iℏ)dδ(x−x′)/dx′. These only make sense in a larger expression in which integration over x′^ is about to be performed.
• The extension to three dimensions is trivial, with the representation of the vector operator pˆ in position space being −iℏ∇. Since differentiation with respect to one coordinate commutes with multiplication by another, [ˆxi, pˆj ] = iℏδij as required. (We have used ˆx as the position operator. We will however use r as the position vector. The reason we don’t use ˆr is that it is so commonly used to mean a unit vector. Thus ˆx|r〉 = r|r〉.) The generalisation of φp(x) is φp(r) = (1/ √ 2 πℏ) eip·r/ℏ, which is a plane wave travelling in the direction of p.
• In the position representation, the Schr¨odinger equation reads

ℏ^2

2 m

∇^2 ψ(r, t) + V (r)ψ(r, t) = iℏ

∂t

ψ(r, t)

Note though that position and time are treated quite differently in quantum mechanics. There is no operator corresponding to time, and t is just part of the label of the state: ψ(r, t) = 〈r|ψ(t)〉.

• Together with the probability density, ρ(r) = |ψ(r)|^2 , we also have a probability flux j(r) = (−iℏ/ 2 m)(ψ(r)∗∇ψ(r) − ψ(r)∇ψ(r)∗). The continuity equation ∇ · j = −∂ρ/∂t which ensures local conservation of probability density follows from the Schr¨odinger equation.

• A two-particle state has a wave function which is a function of the two positions, Φ(r 1 , r 2 ), and the basis kets are direct product states |r 1 〉 ⊗ |r 2 〉. For states of non-interacting distinguishable particles where it is possible to say that the first particle is in single- particle state |ψ〉 and the second in |φ〉, the state of the system is |ψ〉 ⊗ |φ〉 and the wave function is Φ(r 1 , r 2 ) = (〈r 1 | ⊗ 〈r 2 |)(|ψ〉 ⊗ |φ〉) = 〈r 1 |ψ〉〈r 2 |φ〉 = ψ(r 1 )φ(r 2 ).
• From iℏ (^) ddt |ψ(t)〉 = Ĥ |ψ(t)〉 we obtain |ψ(t + dt)〉 =

1 − (^) ℏi Ĥ dt

|ψ(t)〉. Thus

|ψ(t)〉 = lim N →∞

i ℏ

(t − t 0 ) N

H

)N

|ψ(t 0 )〉 = e−i^ Hb(t−t 0 )/ℏ |ψ(t 0 )〉 ≡ U (t, t 0 )|ψ(t 0 )〉

where the exponential of an operator is defined as eΩb^ =

n

1 n!

n. If the Hamiltonian

depends explicitly on time, we have U (t, t 0 ) = T exp

−i

∫ (^) t t 0

H(t

′)dt′/ℏ

, where the time- ordered exponential denoted by T exp means that in expanding the exponential, the op- erators are ordered so that Ĥ (t 1 ) always sits to the right of Ĥ (t 2 ) (so that it acts first) if t 1 < t 2. (This will be derived later, and is given here for completeness.)

References

• Shankar ch 1.
• Mandl ch 12.

1.3 The Stern-Gerlach experiment

Summary: Simple extensions of the Stern-Gerlach experiment reduce the predic- tions about quantum measurement to their essence.

The usual reason for considering the Stern-Gerlach experiment is that it shows experi- mentally that angular momentum is quantised, and that particles can have intrinsic angular momentum which is an integer multiple of 12 ℏ. An inhomogeneous magnetic field deflects par- ticles by an amount proportional to the component of their magnetic moment parallel to the field; when a beam of atoms passes through they follow only a few discrete paths (2j + 1 where j is their total angular momentum quantum number) rather than, as classically predicted, a continuum of paths (corresponding to a magnetic moment which can be at any angle relative to the field). For our purposes though a Stern-Gerlach device is a quick and easily-visualised way of making measurements of quantities (spin components) for which the corresponding operators do not commute, and thereby testing the postulates concerned with measurement. We restrict ourselves to spin-^12 ; all we need to know is that if we write | + n〉 to indicate a particle with its spin up along the direction n, and | − n〉 for spin down, then the two orthogonal states {| ± n〉} span the space for any n, and |〈±n| ± n′〉|^2 = 12 if n and n′^ are perpendicular.

In fact for Ĥ = ˆp

2 2 m +^ V^ (ˆx), we can further show that

d dt

〈xˆ〉 = 〈

pˆ m

〉 and

d dt

〈pˆ〉 = −〈

dV (ˆx) dˆx

which looks very close to Newton’s laws. Note though that 〈dV (ˆx)/dˆx〉 6 = d〈V (ˆx)〉/d〈xˆ〉 in general. This correspondence is not just a coincidence, in the sense that Heisenberg was influenced by it in coming up with his formulation of quantum mechanics. It confirms that it is the expectation value of an operator, rather than the operator itself, which is closer to the classical concept of the time evolution of some quantity as a particle moves along a trajectory. Similarity of formalism is not the same as identity of concepts though. Ehrenfest’s Theorem does not say that the expectation value of a quantity follows a classical trajectory in general. What it does ensure is that if the uncertainty in the quantity is sufficiently small, in other words if ∆x and ∆p are both small (in relative terms) then the quantum motion will aproximate the classical path. Of course because of the uncertainty principle, if ∆x is small then ∆p is large, and it can only be relatively small if p itself is really large—ie if the particle’s mass is macroscopic. More specifically, we can say that we will be in the classical regime if the de Broglie wavelength is much less that the (experimental) uncertainty in x. (In the Stern-Gerlach experiment the atoms are heavy enough that (for a given component of their magnetic moment) they follow approximately classical trajectories through the inhomogeneous magnetic field.)

• Shankar ch 2.7, ch 6
• Mandl ch 3.

Approximate methods I: variational

method and WKB

It is often (almost always!) the case that we cannot solve real problems analytically. Only a very few potentials have analytic solutions, by which I mean one can write down the energy levels and wave functions in closed form, as for the harmonic oscillator and Coulomb potential. In fact those are really the only useful ones (along with square wells)... In the last century, a number of approximate methods have been developed to obtain information about systems which can’t be solved exactly. These days, this might not seem very relevant. Computers can solve differential equations very efficiently. But:

• It is always useful to have a check on numerical methods
• Even supercomputers can’t solve the equations for many interacting particle exactly in a reasonable time (where “many” may be as low as four, depending on the complexity of the interaction) — ask a nuclear physicist or quantum chemist.
• Quantum field theories are systems with infinitely many degrees of freedom. All ap- proaches to QFT must be approximate.
• If the system we are interested in is close to a soluble one, we might obtain more insight from approximate methods than from numerical ones. This is the realm of perturbation theory. The most accurate prediction ever made, for the anomalous magnetic moment of the electron, which is good to one part in 10^12 , is a 4th order perturbative calculation.

2.1 Variational methods: ground state

Summary: Whatever potential we are considering, we can always obtain an upper bound on the ground-state energy.

Suppose we know the Hamiltonian of a bound system but don’t have any idea what the energy of the ground state is, or the wave function. The variational principle states that if we simply guess the wave function, the expectation value of the Hamiltonian in that wave function will be greater than the true ground-state energy:

〈Ψ|Ĥ |Ψ〉 〈Ψ|Ψ〉

≥ E 0

This initially surprising result is more obvious if we consider expanding the (normalised)

|Ψ〉 in the true energy eigenstates |n〉, which gives 〈Ĥ 〉 =

n PnEn. Since all the probabilities

10

• Shankar ch 16.

2.2 Variational methods: excited states

Summary: Symmetry considerations may allow us to extend the variational method to certain excited states.

Looking again at the expression 〈Ĥ 〉 =

n PnEn, and recalling that the^ Pn^ are the squares of the overlap between the trial function and the actual eigenstates of the system, we see that we can only find bounds on excited states if we can arrange for the overlap of the trial wave function with all lower states to be zero. Usually this is not possible. However an exception occurs where the states of the system can be separated into sets with different symmetry properties or other quantum numbers. Examples include parity and (in 3 dimensions) angular momentum. For example the lowest state with odd parity will automatically have zero overlap with the (even-parity) ground state, and so an upper bound can be found for it as well. For the square well, the relevant symmetry is reflection about the midpoint of the well. If we choose a trial function which is antisymmetric about the midpoint, it must have zero overlap with the true ground state. So we can get a good bound on the first excited state, since 〈Ĥ 〉 =

n> 0 PnEn^ > E^1.^ Using Ψ^1 (x) =^ x(a^ −^ x)(2x^ −^ a),^0 < x < a^ we get^ E^1 ≤ 42 ℏ^2 / 2 ma^2 = 1. 064 E 1. If we wanted a bound on E 2 , we’d need a wave function which was orthogonal to both the ground state and the first excited state. The latter is easy by symmetry, but as we don’t know the exact ground state (or so we are pretending!) we can’t ensure the first. We can instead form a trial wave function which is orthogonal to the best trial ground state, but we will no longer have a strict upper bound on the energy E 2 , just a guess as to its value. In this case we can choose Ψ(x) = x(a − x) + bx^2 (a − x)^2 with a new value of b which gives orthogonality to the previous state, and then we get E 2 ∼ 10. 3 E 0 (as opposed to 9 for the actual value).

• Mandl ch 8.
• Shankar ch 16.

2.3 Variational methods: the helium atom

Summary: The most famous example of the variational principle is the ground state of the two-electron helium atom.

If we could switch off the interactions between the electrons, we would know what the ground state of the helium atom would be: Ψ(r 1 , r 2 ) = φZ 100 =2 (r 1 )φZ 100 =2 (r 2 ), where φZnlm is a single-particle wave function of the hydrogenic atom with nuclear charge Z. For the ground state n = 1 and l = m = 0 (spherical symmetry). The energy of the two electrons would be 2 Z^2 ERy = − 108 .8 eV. But the experimental energy is only − 78 .6 eV (ie it takes 78.6 eV to fully ionise neutral helium). The difference is obviously due to the fact that the electrons repel one another.

The full Hamiltonian (ignoring the motion of the proton - a good approximation for the accuracy to which we will be working) is

ℏ^2

2 m

(∇^21 + ∇^22 ) − 2 ℏcα

|r 1 |

|r 2 |

• ℏcα

|r 1 − r 2 |

where ∇^21 involves differentiation with respect to the components of r 1 , and α = e^2 /(4π 0 ℏc) ≈ 1 /137. (See here for a note on units in EM.) A really simple guess at a trial wave function for this problem would just be Ψ(r 1 , r 2 ) as written above. The expectation value of the repulsive interaction term is (5Z/4)ERy giving a total energy of − 74 .8 eV. (Gasiorowicz demonstrates the integral, as do Fitzpatrick and Branson.) It turns out we can do even better if we use the atomic number Z in the wave function Ψ as a variational parameter (that in the Hamiltonian, of course, must be left at 2). The best value turns out to be Z = 27/16 and that gives a better upper bound of − 77 .5 eV – just slightly higher than the experimental value. (Watch the sign – we get an lower bound for the ionization energy.) This effective nuclear charge of less than 2 presumably reflects the fact that to some extent each electron screens the nuclear charge from the other.

• Gasiorowicz ch 14.2, 14.4 (most complete)
• Mandl ch 7.2, 8.8.2 (clearest)
• Shankar ch 16.1 (not very clear; misprints in early printings)

2.4 WKB approximation

Summary: The WKB approximation works for potentials which are slowly- varying on the scale of the wavelength of the particle and is particularly useful for describing tunnelling.

The WKB approximation is named for G. Wentzel, H.A. Kramers, and L. Brillouin, who independently developed the method in 1926. There are pre-quantum antecedents due to Jeffreys and Raleigh, though. We can always write the one-dimensional Schr¨odinger equation as

d^2 φ dx^2

= −k(x)^2 φ(x)

where k(x) ≡

2 m(E − V (x))/ℏ. We could think of the quantity k(x) as a spatially-varying wavenumber (k = 2π/λ), though we anticipate that this can only make sense if it doesn’t change too quickly with position - else we can’t identify a wavelength at all. Let’s see under what conditions a solution of the form

ψ(x) = A exp

±i

∫ (^) x k(x′)dx′

might be a good approximate solution. Plugging this into the SE above, the LHS reads −(k^2 ∓ ik′)ψ. (Here and hereafter, primes denote differentiation wrt x — except when they indicate

For a more general potential, outside the classically allowed region we will have decaying exponentials. In the vicinity of the turning points these solutions will not be valid, but if we approximate the potential as linear we can solve the Schr¨odinger equation exactly (in terms of Airy functions). Matching these to our WKB solutions in the vicinity of x = a and x = b gives the surprisingly simple result that inside the well

ψ(x) =

A

k(x)

cos

(∫ (^) x

a

k(x′)dx′^ − π/ 4

and ψ(x) =

A′

k(x)

cos

(∫ (^) x

b

k(x′)dx′^ + π/ 4

which can only be satisfied if A′^ = ±A and

∫ (^) b a k(x

′)dx′ (^) = (n+ 1 2 )π. This latter is the quantisation condition for a finite well; it is different from the infinite well because the solution can leak into the forbidden region. (For a semi-infinite well, the condition is that the integral equal (n + 34 )π. This is the appropriate form for the l = 0 solutions of a spherically symmetric well.) Unfortunately we can’t check this against the finite square well though, because there the potential is definitely not slowly varying at the edges, nor can it be approximated as linear. But we can try the harmonic oscillator, for which the integral gives Eπ/ℏω and hence the quantisation condition gives E = (n + 12 ) ℏω! The approximation was only valid for large n (small wavelength) but in fact we’ve obtained the exact answer for all levels. Details of the matching process are given in section 2.4.1.1, since I’ve not found them in full detail in any textbook. They are not examinable.

• Gasiorowicz Supplement 4 A
• Shankar ch 16.

2.4.1.1 Matching with Airy Functions

This section is not examinable. More about Airy functions can be found in section A.7. If we can treat the potential as linear over a wide-enough region around the turning points that, at the edges of the region, the WKB approximation is valid, then we can match the WKB and exact solutions. Consider a linear potential V = βx as an approximation to the potential near the right-hand turning point b. We will scale x = ( ℏ^2 /(2mβ))^1 /^3 z and E = (ℏ^2 β^2 / 2 m)^1 /^3 μ, so the turning point is at z = μ. Then the differential equation is y′′(z) − zy(z) + μy(z) = 0 and the solution which decays as z → ∞ is y(z) = AAi(z − μ). This has to be matched, for z not too close to μ, to the WKB solution. In these units, k(x) = (μ−z) and

∫ (^) x b k(x

′)dx′ (^) = ∫^ z μ (μ−z

′)dz′ (^) = −(2/3)(μ−z) 3 / (^2) ,

so

ψWKB x<b (z) =

B

(μ − z)^1 /^4

cos

−^23 (μ − z)^3 /^2 + φ

and ψWKB x>b (z) =

C

(z − μ)^1 /^4

exp

−^23 (z − μ)^3 /^2

(We chose the lower limit of integration to be μ in order that the constant of integration vanished; any other choice would just shift φ.) Now the asymptotic forms of the Airy function are known:

Ai(z) z→−∞ −→

cos

2 3 |z|

3 / (^2) − π 4

π |z|^1 /^4

and Ai(z) z→∞ −→

e−^

(^23) z 3 / 2

2

πz^1 /^4

so

Ai(z − μ) z→−∞ −→

cos

3 (μ^ −^ z)

3 / (^2) − π 4

π(μ − z)^1 /^4

and Ai(z − μ) z→∞ −→

e−^

(^23) (z−μ) 3 / 2

2

π(z − μ)^1 /^4

and these will match the WKB expressions exactly provided C = 2B and φ = π/4. At the left-hand turning point a, the potential is V = −βx (with a different β in general) and the solution y(z) = AAi(−z − μ). On the other hand the WKB integral is

∫ (^) x a k(x

′)dx′ (^) = ∫ (^) z μ (μ^ +^ z

′)dz′ (^) = 2/3(μ + z) 3 / (^2). So in the classically allowed region we are comparing

Ai(−z−μ) z→∞ −→

cos

3 (z^ +^ μ)

3 / (^2) − π 4

π(z + μ)^1 /^4

with ψWKB x>a (z) =

D

(μ + z)^1 /^4

cos

3 (μ^ +^ z)

3 / (^2) + φ)

which requires φ = −π/4. (Note that φ is different in each case because we have taken the integral from a different place.) It is worth stressing that though the exact (Airy function) and WKB solutions match “far away” from the turning point, they do not do so close in. The (z − μ)−^1 /^4 terms in the latter mean that they blow up, but the former are perfectly smooth. They are shown (for μ = 0) below, in red for the WKB and black for the exact functions. We can see they match very well so long as |z − μ| > 1; in fact z → ∞ is overkill!

So now we can be more precise about the conditions under which the matching is possible: we need the potential to be linear over the region ∆x ∼ ( ℏ^2 /(2mβ))^1 /^3 where β = dV /dx. Linearity means that ∆V /∆x ≈ dV /dx at the turning point, or

∣d^2 V /dx^2

∣ (^) ∆x  dV /dx

(assuming the curvature is the dominant non-linearity, as is likely if V is smooth). For the harmonic oscillator,

∣d^2 V /dx^2

∣ (^) ∆x/(dV /dx) = 2(ℏω/E)^2 /^3 which is only much less than 1 for

very large n, making the exact result even more surprising!

2.4.2 WKB approximation for tunnelling

For the WKB approximation to be applicable to tunnelling through a barrier, we need as always |λ′|  1. In practice that means that the barrier function is reasonably smooth and that E  V (x). Now it would of course be possible to do a careful calculation, writing down the WKB wave function in the three regions (left of the barrier, under the barrier and right of the barrier), linearising in the vicinity of the turning points in order to match the wave function and its derivatives at both sides. This however is a tiresomely lengthy task, and we will not attempt it. Instead, recall the result for a high, wide square barrier; the transmission coefficient in the limit e−^2 κ∆L^  1 is given by

T =

16 k 1 k 2 κ^2 (κ^2 + k^21 )(κ^2 + k 22 )

e−^2 κL

where√ k 1 and k 2 are the wavenumbers on either side of the barrier (width L, height V ) and κ = 2 m(V − E). (See the notes for PHYS20101, where however k 1 = k 2 .) All the prefactors are

Data for the lifetimes of long-lived isotopes (those with low-energy alphas) fit such a functional form well, but with 1.61 rather than 1.72. In view of the fairly crude approximations made, this is a pretty good result. Note it is independent of the nuclear radius because we used b  a; we could have kept the first correction, proportional to

b/a, to improve the result.

• Gasiorowicz Supplement 4 B
• Shankar ch 16.

Approximate methods II:

Time-independent perturbation theory

3.1 Formalism

Summary: Perturbation theory is the most widely used approximate method. “Time-independent perturbation theory” deals with bound states eg the spectrum of the real hydrogen atom and its response to extermal fields.

Perturbation theory is applicable when the Hamiltonian Ĥ can be split into two parts, with the first part being exactly solvable and the second part being small in comparison. The first part is always written Ĥ (0), and we will denote its eigenstates by |n(0)〉 and energies by E n(0) (with wave functions φ(0) n ). These we know. The eigenstates and energies of the full Hamiltonian are denoted |n〉 and En, and the aim is to find successively better approximations to these. The zeroth-order approximation is simply |n〉 = |n(0)〉 and En = E n(0) , which is just another way of saying that the perturbation is small. Nomenclature for the perturbing Hamiltonian Ĥ − Ĥ (0)^ varies. δV , Ĥ (1)^ and λĤ (1)^ are all common. It usually is a perturbing potential but we won’t assume so here, so we won’t use the first. The second and third differ in that the third has explicitly identified a small, dimensionless parameter (eg α in EM), so that the residual Ĥ (1)^ isn’t itself small. With the last choice, our expressions for the eigenstates and energies of the full Hamiltonian will be explicitly power series in λ, so En = E n(0) + λE n(1) + λ^2 E n(2) +... etc. With the second choice the small factor is

hidden in Ĥ (1), and is implicit in the expansion which then reads En = E n(0) +E n(1) +E n(2) +.. .. In this case one has to remember that anything with a superscript (1) is first order in this implicit small factor, or more generally the superscript (m) denotes something which is mth order. For the derivation of the equations we will retain an explicit λ, but thereafter we will set it equal to one to revert to the other formulation. We will take λ to be real so that Ĥ 1 is Hermitian. We start with the master equation

( Ĥ (0)^ + λ Ĥ (1))|n〉 = En|n〉.

Then we substitute in En = E n(0) + λE n(1) + λ^2 E n(2) +... and |n〉 = |n(0)〉 + λ|n(1)〉 + λ^2 |n(2)〉 +... and expand. Then since λ is a free parameter, we have to match terms on each side with the same powers of λ, to get

Ĥ (0)|n(0)〉 = E n(0) |n(0)〉 Ĥ (0)|n(1)〉 + Ĥ (1)|n(0)〉 = E n(0) |n(1)〉 + E n(1) |n(0)〉 Ĥ (0)|n(2)〉 + Ĥ (1)|n(1)〉 = E n(0) |n(2)〉 + E n(1) |n(1)〉 + E n(2) |n(0)〉