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7.S: Quantum Mechanics (Summary)

Key Terms

anti-symmetric function odd function
Born interpretation states that the square of a wave function is the probability density
complex function function containing both real and imaginary parts
Copenhagen interpretation states that when an observer is not looking or when a measurement is not being made, the particle has many values of measurable quantities, such as position
correspondence principle in the limit of large energies, the predictions of quantum mechanics agree with the predictions of classical mechanics
energy levels states of definite energy, often represented by horizontal lines in an energy “ladder” diagram
energy quantum number index that labels the allowed energy states
energy-time uncertainty principle energy-time relation for uncertainties in the simultaneous measurements of the energy of a quantum state and of its lifetime
even function in one dimension, a function symmetric with the origin of the coordinate system
expectation value average value of the physical quantity assuming a large number of particles with the same wave function
field emission electron emission from conductor surfaces when a strong external electric field is applied in normal direction to conductor’s surface
ground state energy lowest energy state in the energy spectrum
Heisenberg’s uncertainty principle places limits on what can be known from a simultaneous measurements of position and momentum; states that if the uncertainty on position is small then the uncertainty on momentum is large, and vice versa
infinite square well potential function that is zero in a fixed range and infinitely beyond this range
momentum operator operator that corresponds to the momentum of a particle
nanotechnology technology that is based on manipulation of nanostructures such as molecules or individual atoms to produce nano-devices such as integrated circuits
normalization condition requires that the probability density integrated over the entire physical space results in the number one
odd function in one dimension, a function antisymmetric with the origin of the coordinate system
position operator operator that corresponds to the position of a particle
potential barrier potential function that rises and falls with increasing values of position
principal quantum number energy quantum number
probability density square of the particle’s wave function
quantum dot small region of a semiconductor nanocrystal embedded in another semiconductor nanocrystal, acting as a potential well for electrons
quantum tunneling phenomenon where particles penetrate through a potential energy barrier with a height greater than the total energy of the particles
resonant tunneling tunneling of electrons through a finite-height potential well that occurs only when electron energies match an energy level in the well, occurs in quantum dots
resonant-tunneling diode quantum dot with an applied voltage bias across it
scanning tunneling microscope (STM) device that utilizes quantum-tunneling phenomenon at metallic surfaces to obtain images of nanoscale structures
Schrӧdinger’s time-dependent equation equation in space and time that allows us to determine wave functions of a quantum particle
Schrӧdinger’s time-independent equation equation in space that allows us to determine wave functions of a quantum particle; this wave function must be multiplied by a time-modulation factor to obtain the time-dependent wave function
standing wave state stationary state for which the real and imaginary parts of Ψ(x,t)Ψ(x,t) oscillate up and down like a standing wave (often modeled with sine and cosine functions)
state reduction hypothetical process in which an observed or detected particle “jumps into” a definite state, often described in terms of the collapse of the particle’s wave function
stationary state state for which the probability density function, \(\displaystyle |Ψ(x,t)|^2\), does not vary in time
time-modulation factor factor \(\displaystyle e^{−iωt}\) that multiplies the time-independent wave function when the potential energy of the particle is time independent
transmission probability also called tunneling probability, the probability that a particle will tunnel through a potential barrier
tunnel diode electron tunneling-junction between two different semiconductors
tunneling probability also called transmission probability, the probability that a particle will tunnel through a potential barrier
wave function function that represents the quantum state of a particle (quantum system)
wave function collapse equivalent to state reduction
wave packet superposition of many plane matter waves that can be used to represent a localized particle

Key Equations

Normalization condition in one dimension \(\displaystyle P(x=−∞,+∞)=∫_{−∞}^∞∣Ψ(x,t)∣^2dx=1\)
Probability of finding a particle in a narrow interval of position in one dimension \(\displaystyle (x,x+dx)\) \(\displaystyle P(x,x+dx)=Ψ^∗(x,t)Ψ(x,t)dx\)
Expectation value of position in one dimension \(\displaystyle ⟨x⟩=∫_{−∞}^∞Ψ^∗(x,t)xΨ(x,t)dx\)
Heisenberg’s position-momentum uncertainty principle \(\displaystyle ΔxΔp≥\frac{ℏ}{2}\)
Heisenberg’s energy-time uncertainty principle \(\displaystyle ΔEΔt≥\frac{ℏ}{2}\)
Schrӧdinger’s time-dependent equation \(\displaystyle −\frac{ℏ^2}{2m}\frac{∂^2Ψ(x,t)}{∂x^2}+U(x,t)Ψ(x,t)=iℏ\frac{∂^2Ψ(x,t)}{∂t}\)
General form of the wave function for a time-independent potential in one dimension \(\displaystyle Ψ(x,t)=ψ(x)e^{−iω}\)
Schrӧdinger’s time-independent equation \(\displaystyle −\frac{ℏ^2}{2m}\frac{d^2ψ(x)}{dx^2}+U(x)ψ(x)=Eψ(x)\)
Schrӧdinger’s equation (free particle) \(\displaystyle −\frac{ℏ^2}{2m}\frac{∂^2ψ(x)}{∂x^2}=Eψ(x)\)
Allowed energies (particle in box of length L) \(\displaystyle E_n=n^2\frac{π^2ℏ^2}{2mL^2},n=1,2,3,...\)
Stationary states (particle in a box of length L) \(\displaystyle ψ_n(x)=\sqrt{\frac{2}{L}}sin\frac{nπx}{L},n=1,2,3,...\)
Potential-energy function of a harmonic oscillator \(\displaystyle U(x)=\frac{1}{2}mω^2x^2\)
Stationary Schrӧdinger equation \(\displaystyle −\frac{ℏ}{2m}\frac{d^2ψ(x)}{dx^2}+\frac{1}{2}mω^2x^2ψ(x)=Eψ(x)\)
The energy spectrum \(\displaystyle E_n=(n+\frac{1}{2})ℏω,n=0,1,2,3,...\)
The energy wave functions \(\displaystyle ψ_n(x)=N_ne^{−β^2x^2/2}H_n(βx),n=0,1,2,3,...\)
Potential barrier \(\displaystyle U(x)=\begin{cases}0,& when x<0\\ U_0,& when 0≤x≤L\\0,& whenx>L\end{cases}\)
Definition of the transmission coefficient \(\displaystyle T(L,E)=\frac{|ψ_{tra}(x)|^2}{|ψ_{in}(x)|^2}\)
A parameter in the transmission coefficient \(\displaystyle β^2=\frac{2m}{ℏ^2}(U_0−E)\)
Transmission coefficient, exact \(\displaystyle T(L,E)=\frac{1}{cosh^2βL+(γ/2)^2sinh^2βL}\)
Transmission coefficient, approximate \(\displaystyle T(L,E)=16\frac{E}{U_0}(1−\frac{E}{U_0})e^{−2βL}\)


7.1: Wavefunctions

  • In quantum mechanics, the state of a physical system is represented by a wave function.
  • In Born’s interpretation, the square of the particle’s wave function represents the probability density of finding the particle around a specific location in space.
  • Wave functions must first be normalized before using them to make predictions.
  • The expectation value is the average value of a quantity that requires a wave function and an integration.

7.2: The Heisenberg Uncertainty Principle

  • The Heisenberg uncertainty principle states that it is impossible to simultaneously measure the x-components of position and of momentum of a particle with an arbitrarily high precision. The product of experimental uncertainties is always larger than or equal to \(\displaystyle ℏ/2\).
  • The limitations of this principle have nothing to do with the quality of the experimental apparatus but originate in the wave-like nature of matter.
  • The energy-time uncertainty principle expresses the experimental observation that a quantum state that exists only for a short time cannot have a definite energy.

7.3: The Schrӧdinger Equation

  • The Schrӧdinger equation is the fundamental equation of wave quantum mechanics. It allows us to make predictions about wave functions.
  • When a particle moves in a time-independent potential, a solution of the time-dependent Schrӧdinger equation is a product of a time-independent wave function and a time-modulation factor.
  • The Schrӧdinger equation can be applied to many physical situations.

7.4: The Quantum Particle in a Box

  • Energy states of a quantum particle in a box are found by solving the time-independent Schrӧdinger equation.
  • To solve the time-independent Schrӧdinger equation for a particle in a box and find the stationary states and allowed energies, we require that the wave function terminate at the box wall.
  • Energy states of a particle in a box are quantized and indexed by principal quantum number.
  • The quantum picture differs significantly from the classical picture when a particle is in a low-energy state of a low quantum number.
  • In the limit of high quantum numbers, when the quantum particle is in a highly excited state, the quantum description of a particle in a box coincides with the classical description, in the spirit of Bohr’s correspondence principle.

7.5: The Quantum Harmonic Oscillator

  • The quantum harmonic oscillator is a model built in analogy with the model of a classical harmonic oscillator. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics.
  • The allowed energies of a quantum oscillator are discrete and evenly spaced. The energy spacing is equal to Planck’s energy quantum.
  • The ground state energy is larger than zero. This means that, unlike a classical oscillator, a quantum oscillator is never at rest, even at the bottom of a potential well, and undergoes quantum fluctuations.
  • The stationary states (states of definite energy) have nonzero values also in regions beyond classical turning points. When in the ground state, a quantum oscillator is most likely to be found around the position of the minimum of the potential well, which is the least-likely position for a classical oscillator.
  • For high quantum numbers, the motion of a quantum oscillator becomes more similar to the motion of a classical oscillator, in accordance with Bohr’s correspondence principle.

7.6 The Quantum Tunneling of Particles through Potential Barriers

  • A quantum particle that is incident on a potential barrier of a finite width and height may cross the barrier and appear on its other side. This phenomenon is called ‘quantum tunneling.’ It does not have a classical analog.
  • To find the probability of quantum tunneling, we assume the energy of an incident particle and solve the stationary Schrӧdinger equation to find wave functions inside and outside the barrier. The tunneling probability is a ratio of squared amplitudes of the wave past the barrier to the incident wave.
  • The tunneling probability depends on the energy of the incident particle relative to the height of the barrier and on the width of the barrier. It is strongly affected by the width of the barrier in a nonlinear, exponential way so that a small change in the barrier width causes a disproportionately large change in the transmission probability.
  • Quantum-tunneling phenomena govern radioactive nuclear decays. They are utilized in many modern technologies such as STM and nano-electronics. STM allows us to see individual atoms on metal surfaces. Electron-tunneling devices have revolutionized electronics and allow us to build fast electronic devices of miniature sizes.


Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).