the following, we only briefly describe some essential conclusions. Readers interested in mathematical reasonings and more relevant details are referred to original publications. One-dimensional cases We consider a one-dimensional box with periodic potential of finite length with two ends at and (: potential period, : a positive integer). We are mainly interested in cases where the bulk energy bands have a band gap between two consecutive energy bands. The new theory found that two different types of states exist in such a box with a periodic potential. For each bulk energy band, there are states in the finite crystal whose energies and properties depend on but not and map the energy band exactly. These states are the stationary Bloch states; There is always one and only one state for each band gap, whose energy and properties depend on but not . This state is either a band-edge state or a surface state in the band gap. The very existence of such -dependent states is the fundamental distinction of the quantum confinement of the Bloch waves. Multi-dimensional cases The Schrödinger equation with a multi-dimensional periodic potential is a partial differential equation; the mathematics is more difficult. However, many fundamental understandings can also be obtained. In many essential and straightforward cases, quantum confinement of multi-dimensional Bloch waves in one specific direction between and ( : the period, : the positive integer indicating the box size in the direction) could produce two types of states. Each bulk energy band leads to states whose energy and properties depend on but not and one state whose energy and properties depend on but not . The -dependent states are stationary Bloch states. The energy of the -dependent state is always above the relevant -dependent stationary Bloch states. Again, the very existence of such -dependent states is a fundamental distinction of the quantum confinement of multi-dimensional Bloch waves. For example, in some simplest cases in a simple box of a rectangular cuboid shape having periods separately in three perpendicular dimensions, for each bulk energy band, there are () () () bulk-like states, () () () () () () surface-like states, () () () edge-like states and one vertex-like state. The properties and energy of the vertex-like state depend on three , but neither one ; The properties and energy of each edge-like state depend on two and the other , but neither one of the corresponding two