1.2 The Schrodinger Equation

1.2.1 The Time-Dependent Schrodinger Equation

The wavefunction $\psi(x,t)$ of a non-relativistic particle of mass $m$ obeys the time-dependent Schrodinger equation (TDSE), which is a second-order partial differential equation given by

-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi(x,t)}{\partial x^{2}}+V(x)\psi(x,t% )=i\hbar\frac{\partial\psi(x,t)}{\partial t},

(1.9)

where $\hbar$ is the reduced Planck constant, defined as $\hbar=\frac{h}{2\pi}$ .

The TDSE takes the form

\begin{split}(\text{kinetic energy operator})\times\psi(x,t)&+(\text{potential% energy operator})\times\psi(x,t)=\\ &(\text{total energy operator})\times\psi(x,t).\end{split}

(1.10)

Recall that in classical mechanics, kinetic energy is defined as

K=\frac{1}{2}mv^{2}=\frac{p^{2}}{2m}.

(1.11)

In quantum mechanics, physical quantities are represented by operators, which are linear transformations that act on the wavefunction. Operators obey the same relations as their corresponding quantities in classical mechanics. Hence, the kinetic energy operator in quantum mechanics is defined as

\hat{T}=\frac{\hat{p}^{2}}{2m}=\frac{1}{2m}\left(-i\hbar\frac{\partial}{% \partial x}\right)\left(-i\hbar\frac{\partial}{\partial x}\right)=-\frac{\hbar% ^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}.

(1.12)

Operators act successively to the right, so the product of two derivative operators becomes a second derivative which is then applied to the wavefunction that the operator is acting on.

The total energy operator, or Hamiltonian operator is defined as the sum of the kinetic energy and potential energy operators.

\hat{H}=\hat{T}+\hat{V}=\frac{\hat{p}^{2}}{2m}+\hat{V}=-\frac{\hbar^{2}}{2m}% \frac{\partial^{2}}{\partial x^{2}}+V(x).

(1.13)

The potential energy operator $\hat{V}$ becomes just the function $V(x)$ because the position operator $\hat{x}=x$ . Note that the potential energy function $V(x)$ is often referred to in quantum mechanics as simply “potential”, but this should not be confused with other similarly named functions such as electric potential (voltage, equal to potential energy per unit charge) and gravitational potential (potential energy per unit mass), which are physically different quantities!

The TDSE can be written more compactly using the Hamiltonian as

\hat{H}\psi(x,t)=i\hbar\frac{\partial\psi(x,t)}{\partial t}.

(1.14)

The TDSE is a linear differential equation. This means the the solutions obey the principle of superposition, i.e. if $\psi_{1}$ and $\psi_{2}$ are solutions, then $A\psi_{1}+B\psi_{2}$ is also a solution. This is where the superposition states in quantum mechanics come from.

It is also first-order in time, which implies that specifying the wavefunction at some time $t_{0}$ uniquely specifies the wavefunction for all future times. Thus the wavefunction evolves deterministically according to the Schrodinger equation if the particle is left alone without measurement.

1.2.2 The Time-Independent Schrodinger Equation

We can make one step towards solving the Schrodinger equation in general by separating the time evolution out of the equation. We will assume that the wavefunction $\psi(x,t)$ is the product of a spatial part, that depends only on $x$ , and a temporal part that depends only on $t$ and describes the time evolution.

\psi(x,t)=u(x)T(t).

(1.15)

This technique is known as separation of variables.

If we substitute this into the TDSE (equation 1.9), we get

-\frac{\hbar^{2}}{2m}T(t)\frac{\operatorname{\mathrm{d}}\!^{2}u(x)}{% \operatorname{\mathrm{d}}\!x^{2}}+V(x)u(x)T(t)=i\hbar u(x)\frac{\operatorname{% \mathrm{d}}\!T(t)}{\operatorname{\mathrm{d}}\!t}.

(1.16)

If we divide by $u(x)T(t)$ , we can get all terms depending on $x$ on one side and all terms depending on $t$ on the other:

-\frac{\hbar^{2}}{2m}\frac{1}{u(x)}\frac{\operatorname{\mathrm{d}}\!^{2}u(x)}{% \operatorname{\mathrm{d}}\!x^{2}}+V(x)=\hbar\frac{1}{T(t)}\frac{\operatorname{% \mathrm{d}}\!T(t)}{\operatorname{\mathrm{d}}\!t}.

(1.17)

Now since both sides only depend on a single variable, they must both be constant. This constant must have units of energy, since it must have the same units as $V(x)$ , and it must be equal to the total energy of the particle since that is what is on the left hand side.

If we set the left-hand side equal to the total energy and multiply both sides by $u(x)$ , we get

-\frac{\hbar^{2}}{2m}\frac{\operatorname{\mathrm{d}}\!^{2}u(x)}{\operatorname{% \mathrm{d}}\!x^{2}}+V(x)u(x)=Eu(x).

(1.18)

This equation is of the form

(\text{operator})\times u(x)=(\text{constant})\times u(x),

(1.19)

which is an eigenvalue equation. This means that $u(x)$ represents an eigenfunction of the Hamiltonian operator, with eigenvalue $E$ . This equation is known as the time-independent Schrodinger equation (TISE). Note that the full time dependent separable wavefunction $\psi(x,t)$ is also an eigenfunction of the Hamiltonian, since the spatial derivative does not act on the temporal part. These eigenfunctions, which are usually labelled with some subscript to denote the energy, are known as energy eigenstates.

The TISE can be written more compactly using the Hamiltonian as

\hat{H}\psi=E\psi.

(1.20)

Note that only the energy eigenstates solve the TISE, but any superposition of energy eigenstates can solve the TDSE.

1.2.3 Solving the Temporal Part of the Schrodinger Equation

Looking at the right hand side of equation 1.17, we see that there is nothing unknown, and therefore we are able to solve it in general. Setting the right hand side equal to $E$ and rearranging a bit, we get

\frac{\operatorname{\mathrm{d}}\!T(t)}{\operatorname{\mathrm{d}}\!t}=-\frac{iE% }{\hbar}T(t).

(1.21)

This equation has the solution

T(t)=e^{-\frac{iE}{\hbar}t},

(1.22)

which holds for any potential energy $V(x)$ . This is therefore the temporal part of the energy eigenstates for any problem. It means we do not actually need to solve the full TDSE for every problem we want to study in quantum mechanics. We need only solve the TISE, which is different for every problem because of the presence of the potential, to find the spatial part of the energy eigenstates, and then the full eigenfunctions are given by

\psi(x,t)=u(x)e^{-\frac{iE}{\hbar}t},

(1.23)

where the spatial part $u(x)$ is the eigenfunction of $\hat{H}$ with eigenvalue $E$ .

In particular, the linearity of the TDSE implies that the time-evolution of a superposition of energy eigenstates is equal to the superposition of the time-evolution of those eigenstates. We will use this principle repeatedly to find the most general solutions to the TDSE.